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    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

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Book description

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Reviews

Review of previous edition:'Wow! What a beautifully produced book, and what a wealth of information.'

Don Knuth - Stanford University

Review of previous edition:‘… this lively and exciting volume represents the state of the art in textbooks on computer algebra. Every student and instructor in this area will want a copy.'

Jeffrey O. Shallit Source: Mathematical Reviews

'I find the quality of this book really exceptional …'

Source: Zentralblatt für Mathematik

'This book is a delight: I heartily recommend it.'

Alistair Fitt Source: London Mathematical Society Newsletter

‘On each page I can feel [the authors'] thorough understanding and love for the subject and [their] uncompromising scholarship in presenting this story. A masterpiece.'

Erich Kaltofen - North Carolina State University

'I predict it will be a major success.'

Steve Cook - University of Toronto

‘It's really an impressive work. I'm sure it will become a reference for computer algebra algorithms, and stay it for a long time. Such a reference book was really missing.'

Paul Zimmermann - Universite de Nancy, France

‘I think it's a most successful balance of intuition, rigorous mathematics, interesting and beautiful applications and completeness. The extensive collection of exercises makes it into an ideal textbook for the use in a graduate course.'

Ton Levelt - Radboud University Nijmegen

'I have to say that not only can one apply the book as a flexible textbook for many courses, but its comprehensiveness and clear style also make it an excellent reference text for computer science researchers and grad[uate] students, and anyone else interested in developing exact solution codes.'

Alexander Tzanov Source: Computing Reviews

'… a polished introduction to algorithms for performing algebraic operations on a computer … The book is almost as interesting for the advanced mathematics (mostly in ring and ideal theory and in linear algebra) that is needed to develop the algorithms. It assumes familiarity with the fundamentals of these topics, but does include a 25-page appendix summarizing the needed background. It is well-equipped with exercises, ranging from numerical practice to extensions and variants on results in the body.'

Allen Stenger Source: MAA Reviews

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References
References
John, Abbott, Victor, Shoup, and Paul, Zimmermann (2000), Factorization in ℤ[x]: The Searching Phase. In Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation ISSAC2000, St. Andrews, Scotland, ed. Carlo, Traverso, 1–7. [465]
C. A., AБpamob (1971), О суммировании рациональІΧ функций. Журнаn еычuсnuмеnьноў Мамемамuкu u мамемамuческоu Φuзuκu 11(4), 1071–1075. S. A. Abramov, On the summation of rational functions, U.S.S.R. Computational Mathematics and Mathematical Physics 11(4), 324–330. [671, 675]
S. A., AБpamob (1975), Рациοнальня компοнента решения линейного рекуррентного соотношения первого порядка с рациональной правой частью. Журнаn еычuсnuмеnьноў Μамемамuκu u мамемамuчeckοu Φuзuκu 15(4), 1035–1039. S. A. AБpamov, The rational component of the solution of a first-order linear recurrence relation with rational right side, U.S.S.R. Computational Mathematics and Mathematical Physics 15(4), 216–221. [671]
C. A., AБpamob (1989a), Задачи компьютерной алгебры, связанные с поиском полиномиальных решений линейных дифференциалых и разностных уравнений. Весмнuк Москоєскоѕо Унuєерсмема. Серuя 15. Вычuсnuмеnьная Мамемамuка u Кuбернемка 3, 56–60. S. A. Abpamob, Problems of computer algebra involved in the search for polynomial solutions of linear differential and difference equations, Moscow University Computational Mathematics and Cybernetics 3, 63–68. [641, 671]
C. A., AБpamob (1989b), Рациональные решения линейных дифференциалых и разностных уравнений с полиномиалыми коэффициентами. Журнаn еьlчuсnuмеnbноў Μаmемаmuκu u маmемаmuчeckοu Φuзuκu 29(11), 1611–1620. S. A. Abramov, Rational solutions of linear differential and difference equations with polynomial coefficients, U.S.S.R. Computational Mathematics and Mathematical Physics 29(6), 7–12. [641, 671]
S. A., Abramov (1995), Rational solutions of linear difference and q-difference equations with polynomial coefficients. In Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation ISSAC '95, Montreal, Canada, ed. A. H. M., Levelt, ACM Press, 285–289. [671]
Sergei A., Abramov, Manuel, Bronstein, and Marko, Petkovšek (1995), On Polynomial Solutions of Linear Operator Equations. In Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation ISSAC '95, Montreal, Canada, ed. A. H. M., Levelt, ACM Press, 290–296. [641]
Sergei A., Abramov and Mark, van Hoeij (1999), Integration of solutions of linear functional equations. Integral Transforms and Special Functions 8(1–2), 3–12. [671]
S. A., Abramov and K. Yu., Kvansenko [K. Yu., Kvashenko] (1991), Fast Algorithms to Search for the Rational Solutions of Linear Differential Equations with Polynomial Coefficients. In Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation ISSAC '91,Bonn, Germany, ed. Stephen M., Watt, ACM Press, 267–270. [641]
S. A., Abramov and M., Petkovšek (2001), Canonical Representations of Hypergeometric Terms. In Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe AZ. [675]
L. M., Adleman (1983), On Breaking Generalized Knapsack Public Key Cryptosystems. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, Boston MA, ACM Press, 402–412. [509]
Leonard M., Adleman (1994), Algorithmic Number Theory-The Complexity Contribution. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, Santa Fe NM, ed. Shafi, Goldwasser, IEEE Computer Society Press, Santa Fe NM, 88–113. [531]
Leonard M., Adleman and Hendrik W., Lenstra Jr. (1986), Finding Irreducible Polynomials over Finite Fields. In Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, Berkeley CA, ACM Press, 350–355. [421]
Manindra, Agrawal, Neeraj, Kayal, and Nitin, Saxena (2004), PRIMES is in P. Annals of Mathematics 160(2), 781–793. [517, 543]
Alfred V., Aho, John E., Hopcroft, and Jeffrey D., Ullman (1974), The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading MA. [286, 332]
A. V., Aho, K., Steiglitz, and J. D., Ullman (1975), Evaluating polynomials at fixed sets of points. SIAM Journal on Computing 4, 533–539. [286, 292]
M., Ajtai (1997), The Shortest Vector Problem in L2 is NP-hard for Randomized Reductions. Electronic Colloquium on Computational Complexity TR97-047. 33 pages. [496]
Andres, Albanese, Johannes, Blömer, Jeff, Edmonds, Michael, Luby, and Madhu, Sudan (1994), Priority Encoding Transmission. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, Santa Fe NM, ed. Shafi, Goldwasser, IEEE Computer Society Press, Los Alamitos CA, 604–612. [215]
William Robert, Alford, Andrew, Granville, and Carl, Pomerance (1994), There are infinitely many Carmichael numbers. Annals of Mathematics 140, 703–722. [529, 532]
Gert, Almkvist and Doron, Zeilberger (1990), The Method of Differentiating under the Integral Sign. Journal of Symbolic Computation 10, 571–591. [641, 671]
Noga, Alon, Jeff, Edmonds, and Michael, Luby (1995), Linear Time Erasure Codes With Nearly Optimal Recovery. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, Milwaukee WI, IEEE Computer Society Press, Los Alamitos CA, 512–519. [215]
Francesco, Amoroso (1989), Tests d'appartenance d'après un théorème de Kollár. Comptes Rendus de l'Académie des Sciences Paris, série I 309, 691–694. [618]
George E., Andrews (1994), The Death of Proof? Semi-Rigorous Mathematics?You've Got to Be Kidding! The Mathematical Intelligencer 16(4), 16–18. [697]
Anonymous (1835), Wie sich die Division mit Zahlen erleichtern und zugleich sicherer ausführen läß:t, als auf die gewöhnliche Weise. Journal für die reine und angewandte Mathematik 13(3), 209–218. [41]
Andreas, Antoniou (1979), Digital filters: analysis and design. McGraw-Hill electrical engineering series: Communications and information theory section, McGraw-Hill, New York. [353]
Tom M., Apostol (1983), A Proof that Euler Missed: Evaluating ζ (2) the Easy Way. The Mathematical Intelligencer 5(3), 59–60. Reprinted in Berggren, Borwein & Borwein (1997), 456–457. [62]
Archimedes, (c. 250 BC), Κύκλου μέτρησις (Measurement of a circle). In Opera Omnia, vol. I, ed. I. L., Heiberg, 231–243. B. G. Teubner, Stuttgart, Germany, 1910. Reprinted 1972. [82]
A., Arwin (1918), Über Kongruenzen von dem fünften und höheren Graden nach einem Primzahlmodulus. Arkiv för matematik, astronomi och fysik 14(7), 1–46. [418]
C. A., Asmuth and G. R., Blakley (1982), Pooling, splitting and restituting information to overcome total failure of some channels of communication. In Proceedings 1982 Symposium on Security and Privacy, IEEE Computer Society Press, Los Alamitos CA, 156–159. [131]
A. O. L., Atkin and R. G., Larson (1982), On a primality test of Solovay and Strassen. SIAM Journal on Computing 11(4), 789–791. [532]
L., Babai (1979), Monte Carlo algorithms in graph isomorphism testing. Technical Report 79-10, Département de Mathématique et Statistique, Université de Montréal. [198, 724]
László, Babai, Eugene M., Luks, and Ákos, Seress (1988), Fast Management of Permutation Groups. In Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, White Plains NY, IEEE Computer Society Press, Washington DC, 272–282. [724]
Eric, Bach (1990), Number-theoretic algorithms. Annual Review of Computer Science 4, 119–172. [531]
Eric, Bach (1996), Weil Bounds for Singular Curves. Applicable Algebra in Engineering, Communication and Computing 7, 289–298. [568]
Eric, Bach, Joachim, von zur Gathen, and Hendrik W., Lenstra Jr. (2001), Factoring Polynomials over Special Finite Fields. Finite Fields and Their Applications 7, 5–28. [421]
Eric, Bach, Gary, Miller, and Jeffrey, Shallit (1986), Sums of divisors, perfect numbers and factoring. SIAM Journal on Computing 15(4), 1143–1154. [532, 535]
Eric, Bach and Jeffrey, Shallit (1988), Factoring with cyclotomic polynomials. Mathematics of Computation 52 (185), 201–219. [568]
Eric, Bach and Jeffrey, Shallit (1996), Algorithmic Number Theory, Vol.1: Efficient Algorithms. MIT Press, Cambridge MA. [61, 421, 531, 533, 534]
Eric, Bach and Jonathan, Sorenson (1993), Sieve algorithms for perfect power testing. Algorithmica 9, 313–328. [287]
Eric, Bach and Jonathan, Sorenson (1996), Explicit bounds for primes in residue classes. Mathematics of Computation 65(216), 1717–1735. [529]
Johann Sebastian, Bach (1722), Das Wohltemperierte Klavier. BWV 846–893, Part I appeared in 1722, Part II in 1738. [86]
Claude Gaspar Bachet, de Méziriac (1612), Problèmes plaisans et délectables, qui se font par les nombres. Pierre Rigaud, Lyon. [61]
David H., Bailey, King, Lee, and Horst D., Simon (1990), Using Strassen's Algorithm to Accelerate the Solution of Linear Systems. The Journal of Supercomputing 4(4), 357–371. [2, 337]
George A., Baker Jr. and Peter, Graves-Morris (1996), Padé Approximants. Encyclopedia of Mathematics and its Applications 59, Cambridge University Press, Cambridge, UK, 2nd edition. First edition published in two volumes by Addison-Wesley, Reading MA, 1982. [132]
W. W. Rouse, Ball and H. S. M., Coxeter (1947), Mathematical Recreations & Essays. The Macmillan Company, New York, American edition. First edition 1892. [531, 534]
J. M., Barbour (1948), Music and ternary continued fractions. The American Mathematical Monthly 55, 545–555. [91]
Erwin H., Bareiss (1968), Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination. Mathematics of Computation 22(101–104), 565–578. [132]
Andrej, Bauer and Marko, Petkovšek (1999), Multibasic and Mixed Hypergeometric Gosper-Type Algorithms. Journal of Symbolic Computation 28, 711–736. [671]
Walter, Baur and Volker, Strassen (1983), The complexity of partial derivatives. Theoretical Computer Science 22, 317–330. [352]
David, Bayer and Michael, Stillman (1988), On the complexity of computing syzygies. Journal of Symbolic Computation 6, 135–147. [618]
Paul W., Beame, Russell, Impagliazzo, Jan, Krajíček, Toniann, Pitassi, and Pavel, Pudlák (1996), Lower bounds on Hilbert's Nullstellensatz and propositional proofs. Proceedings of the London Mathematical Society 3, 1–26. [697]
Paul, Beame and Toniann, Pitassi (1998), Propositional Proof Complexity: Past, Present, and Future. Bulletin of the European Association for Theoretical Computer Science 65, 66–89. [697]
Thomas, Becker and Volker, Weispfenning (1993), Gröbner Bases—A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics 141, Springer-Verlag, New York. [618]
Albert H., Beiler (1964), Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover Publications, Inc., New York. [534]
Eric Temple, Bell (1937), Men of Mathematics. Penguin Books Ltd., Harmondsworth, Middlesex. [219, 725, 726, 729]
Christof, Benecke, Roland, Grund, Reinhard, Hohberger, Adalbert, Kerber, Reinhard, Laue, and Thomas, Wieland (1995), MOLGEN, a computer algebra system for the generation of molecular graphs. In Computer Algebra in Science and Engineering,Bielefeld, Germany, August 1994, eds. J., Fleischer, J., Grabmeier, F. W., Hehl, and W., KÜCHLIN, World Scientific, Singapore, 260–272. [698]
M., Ben-Or (1981), Probabilistic algorithms in finite fields. In Proceedings of the 22nd Annual IEEE Symposium on Foundations of Computer Science, Nashville TN, 394–398. [421]
M., Ben-Or, D., Kozen, and J., Reif (1986), The complexity of elementary algebra and geometry. Journal of Computer and System Sciences 32, 251–264. [619]
Michael, Ben-Or and Prasoon, Tiwari (1988), A Deterministic Algorithm For Sparse Multivariate Polynomial Interpolation. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, Chicago IL, ACM Press, 301–309. [498]
Carlos A., Berenstein and Alain, Yger (1990), Bounds for the Degrees in the Division Problem. Michigan Mathematical Journal 37, 25–43. [618]
Lennart, Berggren, Jonathan, Borwein, and Peter, Borwein, eds. (1997), Pi: A Source Book. Springer-Verlag, New York. [90, 729, 735, 737, 749, 751, 753, 761, 763]
E. R., Berlekamp (1967), Factoring polynomials over finite fields. Bell System Technical Journal 46, 1853–1859. [401, 417, 420, 462]
E. R., Berlekamp (1970), Factoring Polynomials Over Large Finite Fields. Mathematics of Computation 24(11), 713–735. [198, 401, 406, 417, 419, 420, 421, 462, 465, 530]
Elwyn R., Berlekamp (1984), Algebraic Coding Theory. Aegean Park Press. First edition McGraw Hill, New York, 1968. [215, 467]
Elwyn R., Berlekamp, Robert J., Mceliece, and Henk C. A., van Tilborg (1978), On the Inherent Intractability of Certain Coding Problems. IEEE Transactions on Information Theory IT-24(3), 384–386. [215]
Benjamin P., Berman and Richard J., Fateman (1994), Optical character recognition for typeset mathematics. In Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation ISSAC '94, Oxford, UK, eds. J. von zur, Gathen and M., Giesbrecht, ACM Press, 348–353. [640]
Joannes, Bernoullius [Johann, Bernoulli] (1703), Problema exhibitum. Acta eruditorum, 26–31. [640]
Daniel J., Bernstein (1998a), Composing Power Series Over a Finite Ring in Essentially Linear Time. Journal of Symbolic Computation 26(3), 339–341. [353]
Daniel J., Bernstein (1998b), Detecting perfect powers in essentially linear time. Mathematics of Computation 67(223), 1253–1283. [287]
Daniel J., Bernstein (2001), Multidigit multiplication for mathematicians. 19 pp. http://cr.yp.to/papers/m3.ps. [247]
P., Bézier (1970), Emploi des Machines à Commande Numérique. Masson & Cie, Paris. English translation: Numerical Control, John Wiley & Sons, 1972. [138]
ÉTienne Bézout (1764), Recherches sur le degré des Équations résultantes de l'évanouissement des inconnues, Et sur les moyens qu'il convient d'employer pour trouver ces Équations. Histoire de l'académie royale des sciences, 288–338. Summary 88–91. [197, 724, 728]
J., Binet (1841), Recherches sur la théorie des nombres entiers et sur la résolution de l'équation indéterminée du premier degré qui n'admet que des solutions entières. Journal de Mathématiques Pures et Appliquées 6, 449–494. [61]
Ian, Blake, Gadiel, Seroussi, and Nigel, Smart (1999), Elliptic Curves in Cryptography. London Mathematical Society Lecture Note Series 265, Cambridge University Press. [580]
Enrico, Bombieri and Alfred J., van der Poorten (1995), Continued fractions of algebraic numbers. In Computational Algebra and Number Theory, eds. Wieb, Bosma and Alf, van der Poorten, Kluwer Academic Publishers, 137–155. [90]
Olaf, Bonorden, Joachim, von zur Gathen, Jürgen, Gerhard, Olaf, Müller, and Michael, Nöcker (2001), Factoring a binary polynomial of degree over one million. ACM SIGSAM Bulletin 35(1), 16–18. [461]
George, Boole (1860), Calculus of finite differences. Chelsea Publishing Company, New York. 5th edition 1970. [669]
A., Borodin and R., Moenck (1974), Fast Modular Transforms. Journal of Computer and System Sciences 8(3), 366–386. [286, 306]
A., Borodin and I., Munro (1975), The Computational Complexity of Algebraic and Numeric Problems. Theory of computation series 1, American Elsevier Publishing Company, New York. [306]
Allan, Borodin and Prasoon, Tiwari (1990), On the Decidability of Sparse Univariate Polynomial Interpolation. In Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, Baltimore MD, ACM Press, 535–545. [498]
J. M., Borwein, P. B., Borwein, and D. H., Bailey (1989), Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi. The American Mathematical Monthly 96(3), 201–219. Reprinted in Berggren, Borwein & Borwein (1997), 623-641. [83]
R. C., Bose and D. K., Ray-Chaudhuri (1960), On A Class of Error Correcting Binary Group Codes. Information and Control 3, 68–79. [215]
Joan, Boyar (1989), Inferring Sequences Produced by Pseudo-Random Number Generators. Journal of the ACM 36(1), 129–141. [505]
Gilles, Brassard and Paul, Bratley (1996), Fundamentals of Algorithmics. Prentice-Hall, Inc., Englewood Cliffs NJ. First published as Algorithmics - Theory & Practice, 1988. [41, 720]
A., Brauer (1939), On addition chains. Bulletin of the American Mathematical Society 45, 736–739.
Richard P., Brent (1976), Analysis of the binary Euclidean algorithm. In Algorithms and Complexity, ed. J. F., Traub, 321–355. Academic Press, New York. [61]
Richard P., Brent (1980), An improved Monte Carlo factorization algorithm. BIT 20, 176–184. [567]
R. P., Brent (1989), Factorization of the eleventh Fermat number (preliminary report). AMS Abstracts 10, 89T-11–73. [542]
Richard P., Brent (1999), Factorization of the tenth Fermat number. Mathematics of Computation 68(225), 429–451. [542, 567]
Richard P., Brent, Fred G., Gustavson, and David Y. Y., Yun (1980), Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants. Journal of Algorithms 1, 259–295. [332]
R. P., Brent and H. T., Kung (1978), Fast Algorithms for Manipulating Formal Power Series. Journal of the ACM 25(4), 581–595. [353, 354]
Richard P., Brent and John M., Pollard (1981), Factorization of the Eighth Fermat Number. Mathematics of Computation 36(154), 627–630. Preliminary announcement in AMS Abstracts 1 (1980), 565. [542, 567]
Ernest F., Brickell (1984), Solving low density knapsacks. In Advances in Cryptology: Proceedings of CRYPTO '83, Plenum Press, New York, 25–37. [509]
Ernest F., Brickell (1985), Breaking iterated knapsacks. In Advances in Cryptology: Proceedings of CRYPTO '84, Santa Barbara, CA. Lecture Notes in Computer Science 196, Springer-Verlag, 342–358. [509]
Egbert, Brieskorn and Horst, Knörrer (1986), Plane Algebraic Curves. Birkhäuser Verlag, Basel. [568]
John, Brillhart, D. H., Lehmer, J. L., Selfridge, Bryant, Tuckerman, and S. S., Wagstaff JR. (1988), Factorizations ofbn ± 1, b = 2,3,5,6,7,10,11,12 up to high powers. Contemporary Mathematics 22, American Mathematical Society, Providence RI, 2nd edition. [542]
Manuel, Bronstein (1990), The Transcendental Risch Differential Equation. Journal of Symbolic Computation 9, 49–60. [641]
Manuel, Bronstein (1991), The Risch Differential Equation on an Algebraic Curve. In Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation ISSAC '91, Bonn, Germany, ed. Stephen M., Watt, ACM Press, 241–246. [641]
Manuel, Bronstein (1992), On solutions of linear ordinary differential equations in their coefficient field. Journal of Symbolic Computation 13, 413–439. [641]
Manuel, Bronstein (1997), Symbolic Integration I—Transcendental Functions. Algorithms and Computation in Mathematics 1, Springer-Verlag, Berlin Heidelberg. [640, 641, 642]
Manuel, Bronstein (2000), On Solutions of Linear Ordinary Difference Equations in their Coefficient Field. Journal of Symbolic Computation 29, 841–877. [671]
Manuel, Bronstein and Anne, Fredet (1999), Solving Linear Ordinary Differential Equations over C(x, e∫ f(x)dx). In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation ISSAC '99, Vancouver, Canada, ed. Sam, Dooley, ACM Press, 173–180. [641]
W. S., Brown (1971), On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors. Journal of the ACM 18(4), 478–504. [62, 197, 198, 199]
W. S., Brown (1978), The Subresultant PRS Algorithm. ACM Transactions on Mathematical Software 4(3), 237–249. [199]
W. S., Brown and J. F., Traub (1971), On Euclid's Algorithm and the Theory of Subresultants. Journal of the ACM 18(4), 505–514. [197, 199, 332]
W. Dale, Brownawell (1987), Bounds for the degrees in the Nullstellensatz. Annals of Mathematics 126, 577–591. [618]
Bruno, Buchberger (1965), Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Philosophische Fakultät an der Leopold-Franzens-Universität, Innsbruck, Austria. [591, 609, 618]
B., Buchberger (1970), Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. aequationes mathematicae 4(3), 271–272 and 374–383. English translation by Michael Abramson and Robert Lumbert in Buchberger & Winkler (1998), 535-545. [618]
B., Buchberger (1976), A theoretical basis for the reduction of polynomials to canonical forms. ACM SIGSAM Bulletin 10(3), 19–29. [618]
B., Buchberger (1985), Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory. In Multidimensional Systems Theory, ed. N. K., Bose, Mathematics and Its Applications, chapter 6, 184–232. D. Reidel Publishing Company, Dordrecht. [618]
Bruno, Buchberger (1987), History and basic features of the critical—pair/completion procedure. Journal of Symbolic Computation 3, 3–38. [618]
Bruno, Buchberger and Franz, Winkler, eds. (1998), Gröbner Bases and Applications. London Mathematical Society Lecture Note Series 251, Cambridge University Press, Cambridge, UK. [618, 738]
James R., Bunch and John E., Hopcroft (1974), Triangular Factorization and Inversion by Fast Matrix Multiplication. Mathematics of Computation 28(125), 231–236. [352]
Peter, Bürgisser (1998), On the Parallel Complexity of the Polynomial Ideal Membership Problem. Journal of Complexity 14, 176–189. [616]
Peter, Bürgisser, Michael, Clausen, and M. Amin, Shokrollahi (1997), Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften 315, Springer-Verlag. [88, 222, 286, 338, 352]
Christoph, Burnikel and Joachim, Ziegler (1998), Fast Recursive Division. Research Report MPI-I-98-1-022, Max-Planck-Institut für Informatik, Saarbrücken, Germany. http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1998-1-022, iv + 27 pages. [286]
S., Buss, R., Impagliazzo, J., Krajíček, P., Pudlák, A. A., Razborov, and J., Sgall (1996/1997), Proof complexity in algebraic systems and bounded depth Frege systems with modular counting. computational complexity 6(3), 256–298. [697]
M. C. R., Butler (1954), On the reducibility of polynomials over a finite field. Quarterly Journal of Mathematics Oxford 5(2), 102–107. [420]
John J., Cade (1987), A modification of a broken public-key cipher. In Advances in Cryptology: Proceedings of CRYPTO '86, Santa Barbara, CA, ed. A. M., Odlyzko. Lecture Notes in Computer Science 263, Springer-Verlag, 64–83. [576]
Paul, Camion (1980), Un algorithme de construction des idempotents primitifs d'idéaux d'algèbres sur Fq. Comptes Rendus de l'Académie des Sciences Paris 291, 479–482. [420]
Paul, Camion (1981), Factorisation des polynômes de Fq. Revue du CETHEDEC 18, 1–17. [419]
Paul, Camion (1982), Un algorithme de construction des idempotents primitifs d'idéaux d'algèbres sur Fq. Annals of Discrete Mathematics 12, 55–63. [419]
Paul F., Camion (1983), Improving an Algorithm for Factoring Polynomials over a Finite Field and Constructing Large Irreducible Polynomials. IEEE Transactions on Information Theory IT-29(3), 378–385. [419]
E. R., Canfield, Paul, Erdős, and Carl, Pomerance (1983), On a problem of Oppenheim concerning ‘Factorisatio Numerorum’. Journal of Number Theory 17, 1–28. [567]
Léandro, Caniglia, André, Galligo, and Joos, Heintz (1988), Borne simple exponentielle pour les degrés dans le théorème des zéros sur un corps de caractéristique quelconque. Comptes Rendus de l'Académie des Sciences Paris, série I 307, 255–258. [619]
Léandro, Caniglia, André, Galligo, and Joos, Heintz (1989), Some new effectivity bounds in computational geometry. In Algebraic Algorithms and Error-Correcting Codes: AAECC-6, Rome, Italy, 1988, ed. T., Mora, Lecture Notes in Computer Science 357, 131–152. Springer-Verlag. [618]
John, Canny (1987), A New Algebraic Method for Robot Motion Planning and Real Geometry. In Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, Los Angeles CA, IEEE Computer Society Press, Washington DC, 39–48. [619]
John F., Canny (1988), The Complexity of Robot Motion Planning. ACM Doctoral Dissertation Award 1987, MIT Press, Cambridge MA. [619]
David G., Cantor (1989), On Arithmetical Algorithms over Finite Fields. Journal of Combinatorial Theory, Series A 50, 285–300. [280, 281, 282, 287]
David G., Cantor and Daniel M., Gordon (2000), Factoring Polynomials over p-Adic Fields. In Algorithmic Number Theory, Fourth International Symposium, ANTS-IV, Leiden, The Netherlands, ed. Wieb, Bosma, Springer-Verlag, 185–208. [466]
David G., Cantor and Erich, Kaltofen (1991), On Fast Multiplication of Polynomials Over Arbitrary Algebras. Acta Informatica 28, 693–701. [245, 247]
David G., Cantor and Hans, Zassenhaus (1981), A New Algorithm for Factoring Polynomials Over Finite Fields. Mathematics of Computation 36(154), 587–592. [405, 406, 417, 418]
Leonard, Carlitz (1932), The arithmetic of polynomials in a Galois field. American Journal of Mathematics 54, 39–50. [426]
R. D., Carmichael (1909/1910), Note on a new number theory function. Bulletin of the American Mathematical Society 16, 232–238. [531]
R. D., Carmichael (1912), On composite numbers P which satisfy the Fermat congruence aP-1 ≡ 1 mod P. The American Mathematical Monthly 19, 22–27. [531]
Thomas R., Caron and Robert D., Silverman (1988), Parallel implementation of the quadratic sieve. The Journal of Supercomputing 1, 273–290. [531, 567]
Paul de Faget, de Casteljau (1985), Shape mathematics and CAD. Hermes Publishing, Paris. [138]
Pietro Antonio, Cataldi (1513), Trattato del modo brevissimo di trouare la radice quadra delli numeri. Bartolomeo Cochi, Bologna. [89]
Augustin, Cauchy (1821), Sur la formule de Lagrange relative à l'interpolation. In Cours d'analyse de l'École Royale Polytechnique (Analyse algébrique), Note V. Imprimerie royale Debure frères, Paris. Œuvres Complètes, IIe série, tome III, Gauthier-Villars, Paris, 1897, 429–433. [132]
Augustin, Cauchy (1840), Mémoire sur l'élimination d'une variable entre deux équations algébriques. In Exercices d'analyse et de physique mathématique, tome 1er. Bachelier, Paris. Œuvres Complètes, IIe série, tome 11. Gauthier-Villars, Paris, 1913, 466–509. [197]
Augustin, Cauchy (1841), Mémoire sur diverses formules relatives à l'Algèbre et à la théorie des nombres. Comptes Rendus de l'Académie des Sciences Paris 12, p. 813 ff. Œuvres Complètes, Ire série, tome 61, Gauthier-Villars, Paris, 1888, 113–146. [131]
Augustin, Cauchy (1847), Mémoire sur les racines des équivalences correspondantes à des modules quelconques premiers ou non premiers, et sur les avantages que présente l'emploi de ces racines dans la théorie des nombres. Comptes Rendus de l'Académie des Sciences Paris 25, p. 37 ff. Œuvres Complètes, Ire série, tome 10, Gauthier-Villars, Paris, 1897, 324–333. [286]
B. F., Caviness (1970), On Canonical Forms and Simplification. Journal of the Association for Computing Machinery 17(2), 385–396. [640]
Arthur, Cayley (1848), On the theory of elimination. The Cambridge and Dublin Mathematical Journal 3, 116–120. Also Cambridge Mathematical Journal 7. [197]
Miguel de Cervantes, Saavedra (1615), El ingenioso cavallero Don Quixote de la Mancha, segunda parte. Francisco de Robles, Madrid. [90]
Jasbir S., Chahal (1995), Manin's Proof of the Hasse Inequality Revisited. Nieuw Archief voor Wiskunde, Vierdeserie 13(2), 219–232. [568]
Bruce W., Char, Keith O., Geddes, and Gaston H., Gonnet (1989), GCDHEU: Heuristic Polynomial GCD Algorithm Based On Integer GCD Computation. Journal of Symbolic Computation 7, 31–48. Extended Abstract in Proceedings of EUROSAM '84, ed. John Fitch, Lecture Notes in Computer Science 174, Springer-Verlag, 285–296. [202]
N., Tschebotareff [N., Chebotarev] (1926), Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören. Mathematische Annalen 95, 191–228. [441, 465]
Π. Л., ЧЕбьІшЕВ (1849), Обь оπредњ леіи числа πростБІχБ чиселБ не πревосхо дяЩихБ данной величинБІ. Mémoires présentés à l'Académie Impériale des sciences de St.-Pétersbourg par divers savants 6, 141–157. P. L. Chebyshev, Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée. Journal de Mathématiques Pures et Appliquées, I série 17 (1852), 341-365. Œuvres I, eds. A. Markoff and N. Sonin, 1899, reprint by Chelsea Publishing Co., New York, 26–48. [533]
P. L., Chebyshev (1852), Mémoire sur les nombres premiers. Journal de Mathématiques Pures et Appliquées, I série 17, 366–390. Mémoires présentées à l'Académie Impériale des sciences de St.-Pétersbourg par divers savants 6 (1854), 17–33. Œuvres I, eds. A. Markoff and N. Sonin, 1899, reprint by Chelsea Publishing Co., New York, 49–70. [533]
Zhi-Zhong, Chen and Ming-Yang, Kao (1997), Reducing Randomness via Irrational Numbers. In Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, El Paso Tx, ACM Press, 200–209. [199]
Zhi-Zhong, Chen and Ming-Yang, Kao (2000), Reducing randomness via irrational numbers. SIAM Journal on Computing 29(4), 1247–1256. [199]
Alexandre L., Chistov (1990), Efficient Factoring Polynomials over Local Fields and Its Applications. In Proceedings of the International Congress of Mathematicians 1990, Kyoto, Japan, vol. II, 1509–1519. Springer-Verlag. [466]
A. L., Chistov and D. Yu., Grigor'ev (1984), Complexity of quantifier elimination in the theory of algebraically closed fields. In Proceedings of the 11th International Symposium Mathematical Foundations of Computer Science 1984, Praha, Czechoslovakia. Lecture Notes in Computer Science 176, Springer-Verlag, Berlin, 17–31. [619]
Benny, Chor and Ronald L., Rivest (1988), A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Transactions on Information Theory IT-34(5), 901–909. Advances in Cryptology: Proceedings of CRYPTO 1984, Santa Barbara CA, Lecture Notes in Computer Science 196, Springer-Verlag, New York, 1985, 54–65. [509]
C.-C., Chou, Y.-F., Deng, G., Li, and Y., Wang (1995), Parallelizing Strassen's Method for Matrix Multiplication on Distributed-Memory MIMD Architectures. Computers & Mathematics with Applications 30(2), 49–69. [352]
Frédéric, Chyzak (1998a), Fonctions holonomes en calcul formel. PhD thesis, École Polytechnique, Paris. [671]
Frédéric, Chyzak (1998b), Gröbner Bases, Symbolic Summation and Symbolic Integration. In Gröbner Bases and Applications, eds. Bruno, Buchberger and Franz, Winkler. London Mathematical Society Lecture Note Series 251, Cambridge University Press, Cambridge, UK, 32–60. [671]
Frédéric, Chyzak (2000), An extension of Zeilberger's fast algorithm to general holonomic functions. Discrete Mathematics 217, 115–134. [671]
Frédéric, Chyzak and Bruno, Salvy (1998), Non-commutative Elimination in Ore Algebras Proves Multivariate Identities. Journal of Symbolic Computation 26(2), 187–227. [671]
Michael, Clausen, Andreas, Dress, Johannes, Grabmeier, and Marek, Karpinski (1991), On Zero-Testing and Interpolation of k-Sparse Multivariate Polynomials over Finite Fields. Theoretical Computer Science 84, 151–164. [498]
Matthew, Clegg, Jeffrey, Edmonds, and Russell, Impagliazzo (1996), Using the Groebner basis algorithm to find proofs of unsatisfiability. In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, Philadelphia PA, ACM Press, 174–183. [679]
G. E., Collins (1966), Polynomial remainder sequences and determinants. The American Mathematical Monthly 73, 708–712. [197, 199]
George E., Collins (1967), Subresultants and Reduced Polynomial Remainder Sequences. Journal of the ACM 14(1), 128–142. [197, 199, 332]
G. E., Collins (1971), The Calculation of Multivariate Polynomial Resultants. Journal of the ACM 18(4), 515–532. [197, 198]
G. E., Collins (1973), Computer algebra of polynomials and rational functions. The American Mathematical Monthly 80, 725–55. [199]
G. E., Collins (1975), Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Lecture Notes in Computer Science 33, Springer-Verlag. [619]
G. E., Collins (1979), Factoring univariate integral polynomials in polynomial average time. In Proceedings of EUROSAM '79,Marseille, France. Lecture Notes in Computer Science 72, 317–329. [455, 465]
George E., Collins and Mark J., Encarnación (1996), Improved Techniques for Factoring Univariate Polynomials. Journal of Symbolic Computation 21, 313–327. [465]
S. A., Cook (1966), On the minimum computation time of functions. Doctoral Thesis, Harvard University, Cambridge MA. [247, 286]
Stephen A., Cook (1971), The Complexity of Theorem-Proving Procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, Shaker Heights OH, ACM Press, 151–158. [722]
James W., Cooley (1987), The Re-Discovery of the Fast Fourier Transform Algorithm. Mikrochimica Acta 3, 33–45. [247, 727]
James W., Cooley (1990), How the FFT Gained Acceptance. In A History of Scientific Computing, ed. Stephen G., Nash, ACM Press, New York, and Addison-Wesley, Reading MA, 133–140. [247]
James W., Cooley and John W., Tukey (1965), An Algorithm for the Machine Calculation of Complex Fourier Series. Mathematics of Computation 19, 297–301. [233, 247]
D., Coppersmith (1993), Solving Linear Equations Over GF(2): Block Lanczos Algorithm. Linear Algebra and its Applications 192, 33–60. [353]
Don, Coppersmith (1994), Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm. Mathematics of Computation 62(205), 333–350. [353]
Don, Coppersmith and Shmuel, Winograd (1990), Matrix Multiplication via Arithmetic Progressions. Journal of Symbolic Computation 9, 251–280. [352, 420]
Robert M., Corless, Erich, Kaltofen, and Stephen M., Watt (2003), Hybrid Methods. In Computer Algebra Handbook - Foundations, Applications, Systems, eds. Johannes, Grabmeier, Erich, Kaltofen, and Volker, Weispfenning, 112–125. Springer-Verlag, Berlin, Heidelberg, New York. [41]
Thomas H., Cormen, Charles E., Leiserson, Ronald L., Rivest, and Clifford, Stein (2009), Introduction to Algorithms. MIT Press, Cambridge MA, London UK, third edition. [41, 368]
James, Cowie, Bruce, Dodson, R. Marije, Elkenbracht-Huizing, Arjen K., Lenstra, Peter L., Montgomery, and Jörg, Zayer (1996), A World Wide Number Field Sieve Factoring Record: On to 512 Bits. In Advances in Cryptology-ASIACRYPT '96. Lecture Notes in Computer Science 1163, Springer-Verlag, 382–394. [569]
David A., Cox (1989), Primes of the Form x2 + ny2 -Fermat, Class Field Theory, and Complex Multiplication. John Wiley & Sons, New York. [568]
David, Cox, John, Little, and Donal, O'shea (1997), Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2nd edition. First edition 1992. [614, 617, 618]
David, Cox, John, Little, and Donal, O'shea (1998), Using Algebraic Geometry. Graduate Texts in Mathematics 185, Springer-Verlag, New York. [617]
Gabriel, Cramer (1750), Introduction a l'analyse des lignes courbes algébriques. Frères Cramer &Cl. Philibert, Genève. [198, 724]
John N., Crossley and Alan S., Henry (1990), Thus Spake al-Khwārizmī: A Translation of the Text of Cambridge University Library Ms. Ii.vi.5. Historia Mathematica 17, 103–131. [727]
Allan J. C., Cunningham and H. J., Woodall (1925), Factorization of (yn 1), y = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers (n). Francis Hodgson, London. [541]
Ivan, Damgård, Peter, Landrock, and Carl, Pomerance (1993), Average case error estimates for the strong probable prime test. Mathematics of Computation 61(203), 177–194. [532]
J. H., Davenport (1986), The Risch differential equation problem. SIAM Journal on Computing 15(4), 903–918. [641]
Pierre, Dèbes (1996), Hilbert subsets and s-integral points. Manuscripta Mathematica 89, 107–137. [498]
Richard A., Demillo and Richard J., Lipton (1978), A probabilistic remark on algebraic program testing. Information Processing Letters 7(4), 193–195. [88, 198]
Angel, Díaz and Erich, Kaltofen (1995), On Computing Greatest Common Divisors with Polynomials Given By Black Boxes for Their Evaluations. In Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation ISSAC '95, Montreal, Canada, ed. A. H. M., Levelt, ACM Press, 232–239. [199]
Angel, Díaz and Erich, Kaltofen (1998), FOXBOX: A System for Manipulating Symbolic Objects in Black Box Representation. In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation ISSAC '98, Rostock, Germany, ed. Oliver, Gloor, ACM Press, 30–37. [498]
Leonard Eugene, Dickson (1919), History of the Theory of Numbers, vol. 1. Carnegie Institute of Washington. Published in 1919, 1920, and 1923 as publication 256. Reprinted by Chelsea Publishing Company, New York, N.Y., 1971. [88]
Whitfield, Diffie and Martin E., Hellman (1976), New directions in cryptography. IEEE Transactions on Information Theory IT-22(6), 644–654. [503, 575, 576, 578, 581]
G. Lejeune, Dirichlet (1837), Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften, 45–81. Werke, Erster Band, ed. L. Kronecker, 1889, 315-342. Reprint by Chelsea Publishing Co., 1969. [528]
G. Lejeune, Dirichlet (1842), Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen. Bericht über die Verhandlungen der Königlich Preussischen Akademie der Wissenschaften, 93–95. Werke, Erster Band, ed. L., Kronecker, 1889, 635-638. Reprint by Chelsea Publishing Co., 1969. [506, 509]
G. Lejeune, Dirichlet (1849), Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften, 69–83. Werke, Zweiter Band, ed. L., Kronecker, 1897, 51-66. Reprint by Chelsea Publishing Co., 1969. [62]
P. G. Lejeune, Dirichlet (1893), Vorlesungen über Zahlentheorie, herausgegeben von R. Dedekind. Friedrich Vieweg & Sohn, Braunschweig, 4th edition. Corrected reprint, Chelsea Publishing Co., New York, 1968. First edition 1863. [707]
John D., Dixon (1970), The Number of Steps in the Euclidean Algorithm. Journal of Number Theory 2, 414–422. [61]
John D., Dixon (1981), Asymptotically Fast Factorization of Integers. Mathematics of Computation 36(153), 255–260. [541, 549, 569]
Bruce, Dodson and Arjen K., Lenstra (1995), NFS with Four Large Primes: An Explosive Experiment. In Advances in Cryptology: Proceedings of CRYPTO '95, Santa Barbara, CA, ed. Don Coppersmith, . Lecture Notes in Computer Science 963, Springer-Verlag, 372–385. [569]
Karl, Dörge (1926), Über die Seltenheit der reduziblen Polynome und der Normalgleichungen. Mathematische Annalen 95, 247–256. [466]
Jean Louis, Dornstetter (1987), On the Equivalence Between Berlekamp's and Euclid's Algorithms. IEEE Transactions on Information Theory IT-33(3), 428–431. [215]
M. W., Drobisch (1855), Über musikalische Tonbestimmung und Temperatur. Abhandlungen der Mathematisch-Physischen Classe der Königlich Sächsischen Gesellschaft der Wissenschaften 4, 1–120 plus 1 table. [91]
Thomas W., Dubé (1990), The structure of polynomial ideals and Gröbner bases. SIAM Journal on Computing 19(4), 750–773. [618]
Raymond, Dubois (1971), Utilisation d'un théorème de Fermat à la découverte des nombres premiers et notes sur les nombres de Fibonacci. Albert Blanchard, Paris. [532]
Lionel, Ducos (2000), Optimizations of the subresultant algorithm. Journal of Pure and Applied Algebra 145, 149–163. [199]
Athanase, Dupré (1846), Sur le nombre des divisions a effectuer pour obtenir le plus grand commun diviseur entre deux nombres entiers. Journal de Mathématiques Pures et Appliquées 11, 41–64. [61]
Wayne, Eberly and Erich, Kaltofen (1997), On Randomized Lanczos Algorithms. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC '97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 176–183. [353]
Jack, Edmonds (1967), Systems of Distinct Representatives and Linear Algebra. Journal of Research of the National Bureau of Standards 71B(4), 241–245. [132]
D., Eisenbud and L., Robbiano, eds. (1993), Computational algebraic geometry and commutative algebra. Symposia Mathematica 34, Cambridge University Press, Cambridge, UK. [617]
D., Eisenbud and B., Sturmfels (1996), Binomial ideals. Duke Mathematical Journal 84(1), 1–45. [697]
G., Eisenstein (1844), Einfacher Algorithmus zur Bestimmung des Werthes von. Journal für die reine und angewandte Mathematik 27(4), 317–318. [533]
Shalosh B., Ekhad (1990), A Very Short Proof of Dixon's Theorem. Journal of Combinatorial Theory, Series A 54, 141–142. [697]
Shalosh B., Ekhad and Sol, Tre (1990), A Purely Verification Proof of the First Rogers-Ramanujan Identity. Journal of Combinatorial Theory, Series A 54, 309–311. [697]
I. Z., Emiris and B., Mourrain (1999), Computer Algebra Methods for Studying and Computing Molecular Conformations. Algorithmica 25(2/3), 372–402. Special Issue on Algorithms for Computational Biology. [698]
Leonhard, Euler (1732/1733), Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus. Commentarii academiae scientiarum imperalis Petropolitanae 6, 103–107. Eneström 26. Opera Omnia, series 1, volume 2, B. G. Teubner, Leipzig, 1915, 1–5. [76, 88, 513, 542]
Leonhard, Euler (1734/1735a), Solutio problematis arithmetici de inveniendo numero qui per datos numeros divisus relinquat data residua. Commentarii academiae scientiarum imperalis Petropolitanae 7, 46–66. Eneström 36. Opera Omnia, series 1, volume 2, B. G. Teubner, Leipzig, 1915, 18–32. [131]
Leonhard, Euler (1734/1735b), De summis serierum reciprocarum. Commentarii Academiae Scientiarum Petropolitanae 7, 123–134. Eneström 41. Opera Omnia, series 1, volume 14, B. G. Teubner, Leipzig, 1925, 73–86. [62]
Leonhard, Euler (1736a), Mechanica sive motus scientia analytice exposita, Tomus I. Typographia Academia Scientiarum, Petropolis. Opera Omnia, series 2, volume 1, B. G. Teubner, Leipzig, 1912. [90]
Leonhard, Euler (1736b), Theorematum quorundam ad numeros primos spectantium demonstratio. Commentarii academiae scientiarum imperalis Petropolitanae 8, 1741, 141–146. Eneström 54. Opera Omnia, series 1, volume 2, B. G. Teubner, Leipzig, 1915, 33-37. [88]
Leonhard, Euler (1737), De fractionibus continuis dissertatio. Commentarii academiae scientiarum imperalis Petropolitanae 9, 1744, 98–137. Eneström 71. Opera Omnia, series 1, volume 14, B. G. Teubner, Leipzig, 1925, 187–215. [89, 90, 91]
Leonhard, Euler (1743), Démonstration de la somme de cette suite 1 + ¼ + + ⅙ + ⅖ + + etc.Journal littéraire d'Allemagne, de Suisse et du Nord (La Haye) 2, 115–127. Bibliotheca Mathematica, Serie 3, 8 1907–1908, 54–60. Eneström 63. Opera Omnia, series 1, volume 14, 177–186. [62]
Leonhard, Euler (1747/1748), Theoremata circa divisores numerorum. Novi commentarii academiae scientiarum imperalis Petropolitanae 1, 20–48. Summarium Novi commentarii academiae scientiarum imperalis Petropolitanae 1, 35–37. Eneström 134. Opera Omnia, series 1, volume 2, B. G. Teubner, Leipzig, 1915, 62–85. [131, 513, 542]
Leonhard, Euler (1748a), Introductio in analysin infinitorum, tomus primus et secundus. M.-M., Bousquet, Lausanne. Opera Omnia, series 1, volume 8 and 9. Teubner, Leipzig, 1922/1945. [62, 90, 132]
Leonhard, Euler (1748b), Sur une contradiction apparente dans la doctrine des lignes courbes. Mémoires de l'Académie des Sciences de Berlin 4, 1750, 219–233. Eneström 147. Opera Omnia, series 1, volume 26, Orell Füssli, Zürich, 1953, 34-45. [198]
Leonhard, Euler (1748c), Démonstration sur le nombre des points où deux lignes des ordres quelconques peuvent se couper. Mémoires de l'Académie des Sciences de Berlin 4, 1750, 234–248. Eneström 148. Opera Omnia, series 1, volume 26, Orell Füssli, Zürich, 1953, 46–59. [197, 198]
Leonhard, Euler (1754/1755), Demonstratio theorematis Fermatiani omnem numerum sive integrum sive fractum esse summam quatuor pauciorumve quadratorum. Novi commentarii academiae scientiarum imperalis Petropolitanae 5, 13–58. Summarium Novi commentarii academiae scientiarum imperalis Petropolitanae 5 6–7. Eneström 242. Opera Omnia, series 1, volume 1, B. G. Teubner, Leipzig, 1915, 339–372. [418]
Leonhard, Euler (1760/1761), Theoremata arithmetica nova methodo demonstrata. Novi commentarii academiae scientiarum imperalis Petropolitanae 8, 74–104. Summarium Novi commentarii academiae scientiarum imperalis Petropolitanae 8, 15–18. Eneström 271. Opera Omnia, series 1, volume 2, B. G. Teubner, Leipzig, 1915, 531–555. [131]
Leonhard, Euler (1761), Theoremata circa residua ex divisione potestatum relicta. Novi commentarii academiae scientiarum imperalis Petropolitanae 7, 49–82. Eneström 262. Opera Omnia, series 1, volume 2, B. G. Teubner, Leipzig, 1915, 493–518. [76, 418]
Leonhard, Euler (1762/1763), Specimen algorithmi singularis. Novi commentarii academiae scientiarum imperalis Petropolitanae 9, 1764, 53–69. Summarium Novi commentarii academiae scientiarum imperalis Petropolitanae 9, 10-13. Eneström 281. Opera Omnia, series 1, volume 15, B. G. Teubner, Leipzig, 1927, 31–49. [90]
Leonhard, Euler (1764), Nouvelle méthode d'éliminer les quantités inconnues des équations. Mémoires de l'Académie des Sciences de Berlin 20, 1766, 91–104. Eneström 310. Opera Omnia, series 1, volume 6, B. G. Teubner, Leipzig, 1921, 197–211. [197, 198]
Leonhard, Euler (1783), De eximio methodi interpolationum in serierum doctrina. Opuscula analytica 1, 157–210. Eneström 555. Opera Omnia, ser. 1, vol. 15, Teubner, Leipzig, 1927, 435–497. [134]
Sergei, Evdokimov (1994), Factorization of Polynomials over Finite Fields in Subexponential Time under GRH. In Algorithmic Number Theory, First International Symposium, ANTS-I, Ithaca, NY, USA. Lecture Notes in Computer Science 877, 209–219. [421]
D. K., Faddeev and V. N., Faddeeva (1963), Computational methods of linear algebra. Freeman, San Francisco, London. Translated by Robert C., Williams. [132]
Robert M., Fano (1949), The transmission of information. Technical Report 65, M.I.T., Research Laboratory of Electronics. [307]
Robert M., Fano (1961), Transmission of information. MIT Press. [307]
J. C., Faugère, P., Gianni, D., Lazard, and T., Mora (1993), Efficient computation of zero-dimensional Gröbner bases by change of ordering. Journal of Symbolic Computation 16, 329–344. [619]
W., Feller (1971), An Introduction to Probability Theory and its Applications. John Wiley & Sons, 2nd edition. [717]
Pierre, Fermat (1636), Letter to Mersenne. In Œuvres de Fermat, vol. 2, Correspondance, eds. Paul, Tannery and Charles, Henry, 63–71. Gauthier-Villars, Paris, 1894. French translation in volume 3, 1894, 286–293. [669]
Charles M., Fiduccia (1972a), Polynomial evaluation via the division algorithm: the fast Fourier transform revisited. In Proceedings of the Fourth Annual ACM Symposium on Theory of Computing, Denver CO, ACM Press, 88–93. [306]
Charles M., Fiduccia (1972b), On obtaining upper bounds on the complexity of matrix multiplication. In Complexity of Computer Computations, eds. Raymond E., Miller and James W., Thatcher, 31–40. Plenum Press, New York. [353]
Charles M., Fiduccia (1973), On the Algebraic Complexity of Matrix Multiplication. PhD thesis, Brown University, Providence RI. [353]
P.-J. E., Finck (1841), Traité élémentaire d'arithmétique à l'usage des candidats aux écoles spéciales. Derivaux, Strasbourg. [61]
Noaï, Fitchas, André, Galligo, and Jacques, Morgenstern (1987), Algorithmes rapides en séquentiel et en parallèle pour l'élimination de quantificateurs en géometrie élémentaire. Séminaire Structures Ordonnées, U. E. R. de Mathématiques, Université de Paris VII. [619]
Noaï, Fitchas, André, Galligo, and Jacques, Morgenstern (1990), Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fields. Journal of Pure and Applied Algebra 67, 1–14. [619]
P., Flajolet, X., Gourdon, and D., Panario (2001), The Complete Analysis of a Polynomial Factorization Algorithm over Finite Fields. Journal of Algorithms 40(1), 37–81. Extended Abstract in Proceedings of the 23rd International Colloquium on Automata, Languages and Programming ICALP 1996, Paderborn, Germany, ed. F. Meyer Auf der Heide and B. Monien, Lecture Notes in Computer Science 1099, Springer-Verlag, 1996, 232-243. [419]
Philippe, Flajolet and Andrew, Odlyzko (1990), Singularity analysis of generating functions. SIAM Journal on Discrete Mathematics 3(2), 216–240. [697]
P., Flajolet, B., Salvy, and P., Zimmermann (1989a), Lambda–Upsilon–Omega: An Assistant Algorithms Analyzer. In Algebraic Algorithms and Error-Correcting Codes: AAECC-6, Rome, Italy, 1988, ed. T., Mora. Lecture Notes in Computer Science 357, Springer-Verlag, 201–212. [697]
Philippe, Flajolet, Bruno, Salvy, and Paul, Zimmermann (1989b), Lambda–Upsilon–Omega—The 1989 CookBook. Rapport de Recherche 1073, INRIA. 116 pages, http://hal.inria.fr/docs/00/07/54/86/PDF/RR-1073.pdf. [697]
Philippe, Flajolet, Bruno, Salvy, and Paul, Zimmermann (1991), Automatic average-case analysis of algorithms. Theoretical Computer Science 79, 37–109. [697]
Menso, Folkerts (1997), Die älteste lateinische Schrift über das indische Rechnen nach al-Hwārizmī. Abhandlungen der Bayerischen Akademie der Wissenschaften, Philosophisch-historische Klasse, neue Folge 113, Verlag der Bayerischen Akademie der Wissenschaften, München. C. H. Beck'sche Verlagsbuchhandlung, München. [286, 727]
J. B. J., Fourier (1822), Théorie Analytique de la Chaleur. Firmin Didot, Paris. [247, 727]
Timothy S., Freeman, Gregory M., Imirzian, Erich, Kaltofen, and Lakshman, Yagati (1988), Dagwood: A System for Manipulating Polynomials Given by Straight-Line Programs. ACM Transactions on Mathematical Software 14(3), 218–240. [498]
Rúsiņš, Freivalds (1977), Probabilistic machines can use less running time. In Information Processing 77—Proceedings of IFIP Congress 77, ed. B., Gilchrist, North-Holland, Amsterdam, 839–842. [88]
Alan M., Frieze, Johan, Håstad, Ravi, Kannan, Jeffrey C., Lagarias, and Adi, Shamir (1988), Reconstructing truncated integer variables satisfying linear congruences. SIAM Journal on Computing 17(2), 262–280. [505]
Ferdinand Georg, Frobenius (1881), Über Relationen zwischen den Näherungsbrüchen von Potenzreihen. Journal für die reine und angewandte Mathematik 90, 1–17. Gesammelte Abhandlungen, Band 2, herausgegeben von J.-P. Serre, Springer-Verlag, Berlin, 1968, 47–63. [132, 197]
G., Frobenius (1896), Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, 689–702. [441, 465]
A., Fröhlich and J. C., Shepherdson (19551956), Effective procedures in field theory. Philosophical Transactions of the Royal Society of London 248, 407–432. [419]
W., Fulton (1969), Algebraic Curves. W. A. Benjamin, Inc., New York. [568]
Martin, Fürer (2009), Faster Integer Multiplication. SIAM Journal on Computing 39(3), 979–1005. [222, 244, 247]
P. X., Gallagher (1973), The large sieve and probabilistic Galois theory. In Analytic Number Theory, ed. Harold G., Diamond. Proceedings of Symposia in Pure Mathematics 24, American Mathematical Society, Providence RI, 91–101. [466]
G., Gallo and B., Mishra (1991), Wu-Ritt Characteristic sets and Their Complexity. In Discrete and Computational Geometry: Papers from the DIMACS Special Year, eds. Jacob E., Goodman, Richard, Pollack, and William, Steiger. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 6, American Mathematical Society and ACM, 111–136. [619]
É., Galois (1830), Sur la théorie des nombres. Bulletin des sciences mathématiques Férussac 13, 428–435. See also Journal de Mathématiques Pures et Appliquées 11 (1846), 398–407, and Écrits et mémoires d'évariste Galois, eds. Robert Bourgne and J.-P. Azra, Gauthier-Villars, Paris, 1962, 112–128. [198, 418, 421, 724, 728]
Taher, Elgamal (1985), A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. IEEE Transactions on Information Theory IT-31(4), 469–472. [580]
Shuhong, Gao (2003), Factoring multivariate polynomials via partial differential equations. Mathematics of Computation 72(242), 801–822. [420]
Shuhong, Gao and Joachim, von zur Gathen (1994), Berlekamp's and Niederreiter's Polynomial Factorization Algorithms. In Finite Fields: Theory, Applications and Algorithms, eds. G. L., Mullen and P. J.-S., Shiue. Contemporary Mathematics 168, American Mathematical Society, 101–115. [420]
Shuhong, Gao, Joachim, von zur Gathen, and Daniel, Panario (1998), Gauss periods: orders and cryptographical applications. Mathematics of Computation 67(221), 343–352. With microfiche supplement. [88, 580]
Shuhong, Gao, Joachim, von zur Gathen, Daniel, Panario, and Victor, Shoup (2000), Algorithms for Exponentiation in Finite Fields. Journal of Symbolic Computation 29(6), 879–889. [88, 580]
Shuhong, Gao and Daniel, Panario (1997), Tests and Constructions of Irreducible Polynomials over Finite Fields. In Foundations of Computational Mathematics, eds. Felipe, Cucker and Michael, Shub, 346–361. Springer Verlag. [419, 421]
Michael R., Garey and David S., Johnson (1979), Computers and intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., San Francisco CA. [509, 722]
Harvey L., Garner (1959), The Residue Number System. IRE Transactions on Electronic Computers, 140–147. [132]
Joachim, von zur Gathen (1984a), Hensel and Newton methods in valuation rings. Mathematics of Computation 42(166), 637–661. [419, 466, 497, 500]
Joachim, von zur Gathen (1984b), Parallel algorithms for algebraic problems. SIAM Journal on Computing 13(4), 802–824. [197, 199]
Joachim, von zur Gathen (1985), Irreducibility of Multivariate Polynomials. Journal of Computer and System Sciences Sciences 31(2), 225–264. [466, 497, 498, 724]
Joachim, von zur Gathen (1986), Representations and parallel computations for rational functions. SIAM Journal on Computing 15(2), 432–452. [131]
Joachim, von zur Gathen (1987), Factoring polynomials and primitive elements for special primes. Theoretical Computer Science 52, 77–89. [421]
Joachim, von zur Gathen (1988), Algebraic complexity theory. Annual Review of Computer Science 3, 317–347. [352]
Joachim, von zur Gathen (1990a), Functional Decomposition of Polynomials: the Tame Case. Journal of Symbolic Computation 9, 281–299. [286, 580, 581]
Joachim, von zur Gathen (1990b), Functional Decomposition of Polynomials: the Wild Case. Journal of Symbolic Computation 10, 437–452. [580]
Joachim, von zur Gathen (1991a), Tests for permutation polynomials. SIAM Journal on Computing 20(3), 591–602. [497]
Joachim, von zur Gathen (1991b), Values of polynomials over finite fields. Bulletin of the Australian Mathematical Society 43, 141–146. [425]
Joachim, von zur Gathen and Jürgen, Gerhard (1996), Arithmetic and Factorization of Polynomials over F2. In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation ISSAC '96, Zürich, Switzerland, ed. Lakshman, Y. N., ACM Press, 1–9. Technical report tr-rsfb-96-018, University of Paderborn, Germany, 1996, 43 pages. Final version in Mathematics of Computation. http://www.juergen-gerhard.net/polyfactTR.ps. [279, 287, 467]
Joachim, von zur Gathen and Jürgen, Gerhard (1997), Fast Algorithms for Taylor Shifts and Certain Difference Equations. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC '97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 40–47. [669, 670]
Joachim, von zur Gathen and Silke, Hartlieb (1998), Factoring Modular Polynomials. Journal of Symbolic Computation 26(5), 583–606.
Extended abstract in Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation ISSAC '96, Zürich, Switzerland, 10–17. [466]
Joachim, von zur Gathen and Erich, Kaltofen (1985), Factoring Sparse Multivariate Polynomials. Journal of Computer and System Sciences 31(2), 265–287. [497]
Joachim, von zur Gathen, Marek, Karpinski, and Igor E., Shparlinski (1996), Counting curves and their projections. computational complexity 6, 64–99. Extended abstract in Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing, San Diego CA (1993), 805–812. [198]
Joachim, von zur Gathen and Thomas, Lücking (2000), Subresultants revisited. In Proceedings of LATIN 2000, Punta del Este, Uruguay, eds. Gastón H. Gonnet, Daniel, Panario, and Alfredo, Viola. Lecture Notes in Computer Science 1776, Springer-Verlag, 318–342. Final version in von zur Gathen & Lücking (2003). [197]
Joachim, von zur Gathen and Thomas, Lücking (2003), Subresultants revisited. Theoretical Computer Science 297, 199–239. [199, 746]
Joachim, von zur Gathen and Michael, Nöcker (1997), Exponentiation in Finite Fields: Theory and Practice. In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: AAECC-12, Toulouse, France, eds. Teo, Mora and Harold, Mattson. Lecture Notes in Computer Science 1255, Springer-Verlag, 88–113. [88, 580]
Joachim, von zur Gathen and Michael, Nöcker (1999), Computing Special Powers in Finite Fields: Extended Abstract. In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation ISSAC '99, Vancouver, Canada, ed. Sam, Dooley, ACM Press, 83–90. [88]
Joachim, von zur Gathen and Daniel, Panario (2001), Factoring Polynomials Over Finite Fields: A Survey. Journal of Symbolic Computation 31(1–2), 3–17. [419]
Joachim, von zur Gathen and Victor, Shoup (1992), Computing Frobenius maps and factoring polynomials. computational complexity 2, 187–224. [405, 406, 419, 420, 467]
Joachim, von zur Gathen and Igor E., Shparlinski (2006), GCD of Random Linear Combinations. Algorithmica 46(1), 137–148. [199]
Carl Friedrich, Gauss (1801), Disquisitiones Arithmeticae. Gerh. Fleischer Iun., Leipzig. English translation by Arthur A., Clarke, Springer-Verlag, New York, 1986. [131, 197, 372, 497, 728]
Carl Friedrich, Gauss (1809), Theoria motus corporum coelestium in sectionibus conicis solem ambientum. Perthes und Besser, Hamburg. Werke VII, Königliche Gesellschaft der Wissenschaften, Göttingen, 1906, 1–288. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. [131]
Carl Friedrich, Gauss (1810), Disquisitio de elementis ellipticis Palladis ex oppositionibus annorum 1803, 1804, 1805, 1807, 1808, 1809. Commentationes societatis regiae scientarium Gottingensis recentiores 1(1811), 3–24. Werke VI, Königliche Gesellschaft der Wissenschaften, Göttingen, 1874, 3–24. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. Announcement in Göttingische gelehrte Anzeigen (1810), Werke VI, 1874, 61–64. [131]
Carl Friedrich, Gauss (1831), Theoria residuorum biquadraticorum, commentatio secunda. Commentationes societatis regiae scientiarum Gottingensis recentiores 7(1832). Werke II, Königliche Gesellschaft der Wissenschaften, Göttingen, 1863, 93–148. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. Announcement in Göttingische gelehrte Anzeigen (1831), Werke II, 1863, 169–178. [724]
Carl Friedrich, Gauss (1849), Brief an Encke, 24. Dezember 1849. In Werke II, Handschriftlicher Nachlass, 444–447. Königliche Gesellschaft der Wissenschaften, Göttingen, 1863. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. [533]
Carl Friedrich, Gauss (1863a), Solutio congruentiae Xm − 1 ≡ 0. Analysis residuorum. Caput sextum. Pars prior. In Werke II, Handschriftlicher Nachlass, ed. R., Dedekind, 199–211. Königliche Gesellschaft der Wissenschaften, Göttingen. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. [417, 466, 724]
Carl Friedrich, Gauss (1863b), Disquisitiones generales de congruentiis. Analysis residuorum caput octavum. In Werke II, Handschriftlicher Nachlass, ed. R., Dedekind, 212–240. Königliche Gesellschaft der Wissenschaften, Göttingen. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. [62, 197, 373, 417, 418, 419, 421, 466, 729]
Carl Friedrich, Gauss (1863c), Zur Theorie der complexen Zahlen. In Werke II, Handschriftlicher Nachlass, 387–398. Königliche Gesellschaft der Wissenschaften, Göttingen. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. [724]
Carl Friedrich, Gauss (1866), Theoria interpolationis methodo nova tractata. In Werke III, Nachlass, 265–330. Königliche Gesellschaft der Wissenschaften, Göttingen. Reprinted by Georg Olms Verlag, Hildesheim New York, 1973. [90, 247]
Leopold, Gegenbauer (1884), Asymptotische Gesetze der Zahlentheorie. Denkschriften der kaiserlichen Akademie der Wissenschaften Wien 49, 37–80. [62]
W. M., Gentleman and G., Sande (1966), Fast Fourier transforms—for fun and profit. In Proceedings of the Fall Joint Computer Conference, San Francisco CA. AFIPS Conference Proceedings 29, Spartan books, Washington DC, 563–578. [247]
François, Genuys (1958), Dix mille décimales de π. Chiffres 1, 17–22. [82]
Joseph Diaz, Gergonne (1822), De la recherche des facteurs rationnels des polynomes. Annales de mathématiques pures et appliquées 12, 309–316. [465]
Jürgen, Gerhard (1998), High degree solutions of low degree equations. In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation ISSAC '98, Rostock, Germany, ed. Oliver, Gloor, ACM Press, 284–289. [674]
Jürgen, Gerhard (2001a), Fast Modular Algorithms for Squarefree Factorization and Hermite Integration. Applicable Algebra in Engineering, Communication and Computing 11(3), 203–226. [470, 640]
Jürgen, Gerhard (2001b), Modular algorithms in symbolic summation and symbolic integration. Lecture Notes in Computer Science 3218, Springer-Verlag, Berlin, Heidelberg. [641, 670]
J., Gerhard, M., Giesbrecht, A., Storjohann, and E. V., Zima (2003), Shiftless Decomposition and Polynomial-time Rational Summation. In Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation ISSAC2003, Philadelphia PA, ed. J. R., Sendra, ACM Press, 119–126. [671]
M., Giesbrecht, A., Lobo, and B. D., Saunders (1998), Certifying Inconsistency of Sparse Linear Systems. In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation ISSAC '98, Rostock, Germany, ed. Oliver, Gloor, ACM Press, 113–119. [353]
Mark, Giesbrecht, Arne, Storjohann, and Gilles, Villard (2003), Algorithms for Matrix Canonical Forms. In Computer Algebra Handbook – Foundations, Applications, Systems, eds. Johannes, Grabmeier, Erich, Kaltofen, and Volker, Weispfenning, 38–41. Springer-Verlag, Berlin, Heidelberg, New York. [353]
John, Gill (1977), Computational complexity of probabilistic Turing machines. SIAM Journal on Computing 6(4), 675–695. [198]
Alessandro, Giovini, Teo, Mora, Gianfranco, Niesi, Lorenzo, Robbiano, and Carlo, Traverso (1991), “One sugar cube, please” or Selection strategies in the Buchberger algorithm. In Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation ISSAC '91, Bonn, Germany, ed. Stephen M., Watt, ACM Press, 49–54. [619]
M., Giusti (1984), Some effectivity problems in polynomial ideal theory. In Proceedings of EUROSAM '84, Cambridge, UK, ed. John, Fitch, Lecture Notes in Computer Science 174, 159–171. Springer-Verlag, Berlin. [618]
Marc, Giusti and Joos, Heintz (1991), Algorithmes – disons rapides – pour la décomposition d'une variété algébrique en composantes irréductibles et équidimensionnelles. In Proceedings of Effective Methods in Algebraic Geometry MEGA '90, eds. Teo, Mora and Carlo, Traverso. Progress in Mathematics 94, Birkhäuser Verlag, Basel, 169–193. [619]
Nobuhiro, and Harold A., Scheraga (1970), Ring Closure and Local Conformational Deformations of Chain Molecules. Macromolecules 3(2), 178–187. [698]
Hermann H., Goldstine (1977), A History of Numerical Analysis from the 16th through the 19th Century. Studies in the History of Mathematics and Physical Sciences 2, Springer-Verlag, New York. [286]
R. M. F., Goodman and A. J., McAuley (1984), A New Trapdoor Knapsack Public Key Cryptosystem. In Advances in Cryptology: Proceedings of EUROCRYPT 1984, Paris, France, eds. T., Beth, N., Cot, and I., Ingemarsson. Lecture Notes in Computer Science 209, Springer-Verlag, Berlin, 150–158. [509]
Paul, Gordan (1885), Vorlesungen über Invariantentheorie. Erster Band: Determinanten. B. G. Teubner, Leipzig. Herausgegeben von Georg Kerschensteiner. [199, 332]
Daniel M., Gordon (1993), Discrete logarithms in GF(p) using the number field sieve. SIAM Journal on Discrete Mathematics 6(1), 124–138. [579]
R. William, Gosper Jr. (1978), Decision procedure for indefinite hypergeometric summation. Proceedings of the National Academy of Sciences of the USA 75(1), 40–42. [641, 662, 670, 671, 675]
R., Göttfert (1994), An acceleration of the Niederreiter factorization algorithm in characteristic 2. Mathematics of Computation 62(206), 831–839. [420]
Xavier, Gourdon (1996), Combinatoire, Algorithmique et Géométrie des Polynômes. PhD thesis, École Polytechnique, Paris. [419]
R. L., Graham, D. E., Knuth, and O., Patashnik (1994), Concrete Mathematics. Addison-Wesley, Reading MA, 2nd edition. First edition 1989. [571, 669, 670, 717, 720]
J. P., Gram (1883), Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate. Journal für die reine und angewandte Mathematik 94, 41–73. [496]
Andrew, Granville (1990), Bounding the Coefficients of a Divisor of a Given Polynomial. Monatshefte für Mathematik 109, 271–277. [198]
D. Yu., Grigor'ev (1988), Complexity of deciding Tarski algebra. Journal of Symbolic Computation 4(1/2). [619]
Dima Yu., Grigoriev, Marek, Karpinski, and Michael F., Singer (1990), Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal on Computing 19(6), 1059–1063. [498]
Dima, Grigoriev, Marek, Karpinski, and Michael F., Singer (1994), Computational complexity of sparse rational interpolation. SIAM Journal on Computing 23(1), 1–11. [498]
H. F., de Groote (1987), Lectures on the Complexity of Bilinear Problems. Lecture Notes in Computer Science 245, Springer-Verlag. [352]
Martin, Grötschel, László, Lovász, and Alexander, Schrijver (1993), Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics 2, Springer-Verlag, Berlin, Heidelberg, 2nd edition. First edition 1988. [496]
L. J., Guibas and A. M., Odlyzko (1980), Long Repetitive Patterns in Random Sequences. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 53, 241–262. [205]
Richard K., Guy (1975), How to factor a number. In Proceedings of the Fifth Manitoba Conference on Numerical Mathematics, 49–89. [568, 569]
Walter, Habicht (1948), Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens. Commentarii Mathematici Helvetici 21, 99–116. [199]
J., Hadamard (1893), Résolution d'une question relative aux déterminants. Bulletin des Sciences Mathématiques 17, 240–246. [496]
J., Hadamard (1896), Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bulletin de la Société mathématique de France 24, 199–220. [533]
Armin, Haken (1985), The intractability of resolution. Theoretical Computer Science 39, 297–308. [678]
Paul R., Halmos (1985), I want to be a mathematician. Springer-Verlag. [533]
John H., Halton (1970), A retrospective and prospective survey of the Monte Carlo method. SIAM Review 12(1), 1–63. [198]
Richard W., Hamming (1986), Coding and Information Theory. Prentice-Hall, Inc., Englewood Cliffs NJ, 2nd edition. First edition 1980. [308]
G. H., Hardy (1937), The Indian Mathematician Ramanujan. The American Mathematical Monthly 44, 137–155. Collected Papers, volume VII, Clarendon Press, Oxford, 1979, 612–630. [535]
Godfrey Harold, Hardy (1940), A mathematician's apology. Cambridge University Press, Cambridge, UK. [26, 726, 728]
G. H., Hardy and E. M., Wright (1985), An introduction to the theory of numbers. Clarendon Press, Oxford, 5th edition. First edition 1938. [62, 421, 532, 534]
William, Hart, Mark, van Hoeij, and Andrew, Novocin (2011), Practical Polynomial Factoring in Polynomial Time. In Proceedings of the 2011 International Symposium on Symbolic and Algebraic Computation ISSAC2011, San Jose CA, ed. Anton, Leykin, ACM Press, 163–170. [497]
Robin, Hartshorne (1977), Algebraic Geometry. Graduate Texts in Mathematics 52, Springer-Verlag, New York. [568]
M. W., Haskell (1891/1892), Note on resultants. Bulletin of the New York Mathematical Society 1, 223–224. [332]
Helmut, Hasse (1933), Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F. K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen Fällen. Vorläufige Mitteilung.
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 42, 253–262. [568]
Johan, Håstad and Mats, Näslund (1998), The Security of Individual RSA Bits. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, Palo Alto CA, IEEE Computer Society Press, Los Alamitos CA, 510–519. [580]
Timothy F., Havel and Igor, Najfeld (1995), A new system of equations, based on geometric algebra, for the ring closure in cyclic molecules. In Computer Algebra in Science and Engineering, Bielefeld, Germany, August 1994, eds. J., Fleischer, J., Grabmeier, F. W., Hehl, and W., Küchlin, World Scientific, Singapore, 243–259. [698]
P., Hazebroek and L. J., Oosterhoff (1951), The isomers of cyclohexane. Discussions of the Faraday Society 10, 88–93. [698]
Thomas L., Heath, ed. (1925), The thirteen books of Euclid's elements, vol. 1. Dover Publications, Inc., New York, Second edition. First edition appeared 1908. Translated from the text of Heiberg. [24, 25]
Michael T., Heideman, Don H., Johnson, and C. Sidney, Burrus (1984), Gauss and the history of the Fast Fourier Transform. IEEE ASSP Magazine, 14–21. [247]
H., Heilbronn (1968), On the average length of a class of finite continued fractions. In Abhandlungen aus Zahlentheorie und Analysis. Zur Erinnerung an Edmund Landau (1877–1938), ed. Paul, Túran, 87–96. VEB Deutscher Verlag der Wissenschaften, Berlin. Also in Number Theory and Analysis, a Collection of Papers in Honor of Edmund Landau (1877–1938), Plenum Press, New York, 1969. [61]
Joos, Heintz, Tomas, Recio, and Marie-Françoise, Roy (1991), Algorithms in Real Algebraic Geometry and Applications to Computational Geometry. In Discrete and Computational Geometry: Papers from the DIMACS Special Year, eds. Jacob E., Goodman, Richard, Pollack, and William, Steiger. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 6, American Mathematical Society and ACM, 137–163. [619]
Joos, Heintz and Malte, Sieveking (1981), Absolute Primality of Polynomials is Decidable in Random Polynomial Time in the Number of Variables. In Proceedings of the 8th International Colloquium on Automata, Languages and Programming ICALP 1981, Acre ('Akko), Israel. Lecture Notes in Computer Science 115, Springer-Verlag, 16–27. [497]
Peter A., Hendriks and Michael F., Singer (1999), Solving Difference Equations in Finite Terms. Journal of Symbolic Computation 27, 239–259. [671]
Kurt, Hensel (1918), Eine neue Theorie der algebraischen Zahlen. Mathematische Zeitschrift 2, 433–452. [444, 466]
Grete, Hermann (1926), Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Mathematische Annalen 95, 736–788. [616]
C., Hermite (1872), Sur l'intégration des fractions rationnelles. Annales de Mathématiques, 2ème série 11, 145–148. [640]
Nicholas J., Higham (1990), Exploiting Fast Matrix Multiplication Within the Level 3 BLAS. ACM Transactions on Mathematical Software 16(4), 352–368. [337]
David, Hilbert (1890), Ueber die Theorie der algebraischen Formen. Mathematische Annalen 36, 473–534. [586, 616, 618]
David, Hilbert (1892), Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten. Journal für die reine und angewandte Mathematik 110, 104–129. [495, 586]
David, Hilbert (1893), Ueber die Transcendenz der Zahlen e und π. Mathematische Annalen 43, 216–219. Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen 2 (1893), 113–116. Reprinted in Berggren, Borwein & Borwein (1997), 226–229. [90]
David, Hilbert (1900), Mathematische Probleme. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 253–297. Archiv für Mathematik und Physik, 3. Reihe 1 (1901), 44–63 and 213–237. English translation: Mathematical Problems, Bulletin of the American Mathematical Society 8 (1902), 437–479. [587, 726]
David, Hilbert (1930), Probleme der Grundlegung der Mathematik. Mathematische Annalen 102, 1–9. [419]
Heisuke, Hironaka (1964), Resolution of singularities of an algebraic variety over a field of characteristic zero. Annals of Mathematics 79(1), I: 109–203, II: 205–326. [591]
A., Hocquenghem (1959), Codes correcteurs d'erreurs. Chiffres 2, 147–156. [215]
Mark, van Hoeij (1998), Rational Solutions of Linear Difference Equations. In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation ISSAC '98, Rostock, Germany, ed. Oliver, Gloor, ACM Press, 120–123. [671]
Mark, van Hoeij (1999), Finite singularities and hypergeometric solutions of linear recurrence equations. Journal of Pure and Applied Algebra 139, 109–131. [671]
Mark, van Hoeij (2002), Factoring polynomials and the knapsack problem. Journal of Number Theory 96(2), 167–189. [497]
Joris, van der Hoeven (1997), Lazy Multiplication of Formal Power Series. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC '97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 17–20. [469]
C. M., Hoffman, J. R., Sendra, and F., Winkler, eds. (1997), Parametric Algebraic Curves and Applications. Special Issue of the Journal of Symbolic Computation 23(2/3). [618]
D. G., Hoffman, D. A., Leonard, C. C., Lindner, K. T., Phelps, C. A., Rodger, and J. R., Wall (1991), Coding Theory: The Essentials. Marcel Dekker, Inc., New York. [215]
Ellis, Horowitz (1971), Algorithms for partial fraction decomposition and rational function integration. In Proceedings 2nd ACM Symposium on Symbolic and Algebraic Manipulation, Los Angeles CA, ed. S. R., Petrick, ACM Press, 441–457. [627]
Ellis, Horowitz (1972), A fast method for interpolation using preconditioning. Information Processing Letters 1, 157–163. [306]
Jeremy, Horwitz and Ramarathnam, Venkatesan (2002), Random Cayley Digraphs and the Discrete Logarithm. In Algorithmic Number Theory Symposium V, ANTS-V, eds. Claus, Fieker and David R., Kohel. Lecture Notes in Computer Science 2369, Springer-Verlag, 100–114. [567]
Ming-Deh A., Huang (1985), Riemann Hypothesis and Finding Roots over Finite Fields. In Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, Providence RI, ACM Press, 121–130. [421]
Ming-Deh, Huang and Yiu-Chung, Wong (1998), Extended Hilbert Irreducibility and its Applications. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms SODA '98, 50–58. [498]
Xiaohan, Huang and Victor Y., Pan (1998), Fast Rectangular Matrix Multiplication and Applications. Journal of Complexity 14, 257–299. [353, 405, 420]
David A., Huffman (1952), A Method for the Construction of Minimum-Redundancy Codes. Proceedings of the I.R.E. 40(9), 1098–1101. [307, 368]
Christianus, Hugenius [Christiaan, Huygens] (1703), Descriptio Automati Planetarii. In Opuscula postuma, quae continent: Dioptricam. Commentarios de vitris figurandis. Dissertationem de corona & parheliis. Tractatum de motu/de vi centrifuga. Descriptionem automati planetarii. Cornelius Boutesteyn, Leyden. [89]
Thomas W., Hungerford (1990), Abstract Algebra: An Introduction. Saunders College Publishing, Philadelphia PA. [703]
A., Hurwitz (1891), Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Mathematische Annalen 39, 279–284. [90]
Dung T., Huynh (1986), A Superexponential Lower Bound for Gröbner Bases and Church-Rosser Commutative Thue Systems. Information and Control 68(1–3), 196–206. [618]
C. G. J., Jacobi (1836), De eliminatione variabilis e duabus aequationibus algebraicis. Journal für die reine und angewandte Mathematik 15, 101–124. [197]
C. G. J., Jacobi (1846), Über die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function. Journal für die reine und angewandte Mathematik 30, 127–156. [132, 197]
C. G. J., Jacobi (1868), Allgemeine Theorie der kettenbruchähnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird. Journal für die reine und angewandte Mathematik 69, 29–64. [91]
Tudor, Jebelean (1997), Practical Integer Division with Karatsuba Complexity. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC '97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 339–341. [286]
David S., Johnson (1990), A Catalog of Complexity Classes. In Handbook of Theoretical Computer Science, vol. A, ed. J., van Leeuwen, 67–161. Elsevier Science Publishers B.V., Amsterdam, and The MIT Press, Cambridge MA. [724]
William, Jones (1706), Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics, London. [90]
Charles, Jordan (1965), Calculus of finite differences. Chelsea Publishing Company, New York. First edition Röttig and Romwalter, Sopron, Hungary, 1939. [669]
Norbert, Kajler and Neil, Soiffer (1998), A Survey of User Interfaces for Computer Algebra Systems. Journal of Symbolic Computation 25, 127–159. [21]
K., Kalorkoti (1993), Inverting polynomials and formal power series. SIAM Journal on Computing 22(3), 552–559. [286]
E., Kaltofen (1982), Factorization of Polynomials. In Computer Algebra, Symbolic and Algebraic Computation, eds. B., Buchberger, G. E., Collins, and R., Loos, 95–113. Springer-Verlag, New York, 2nd edition. [419]
Erich, Kaltofen (1983), On the Complexity of Finding Short Vectors in Integer Lattices. In Proceedings of EUROCAL 1983, London, UK. Lecture Notes in Computer Science 162, Springer-Verlag, Berlin / New York, 236–244. [497]
Erich, Kaltofen (1984), A Note on the Risch Differential Equation. In Proceedings of EUROSAM '84, Cambridge, UK, ed. John, Fitch. Lecture Notes in Computer Science 174, Springer-Verlag, Berlin, 359–366. [641]
Erich, Kaltofen (1985a), Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM Journal on Computing 14(2), 469–489. [497]
Erich, Kaltofen (1985b), Effective Hilbert Irreducibility. Journal of Computer and System Sciences 66, 123–137. [498]
E., Kaltofen (1989), Factorization of Polynomials Given by Straight-Line Programs. In Randomness and Computation, ed. S., Micali, JAI Press, Greenwich CT, 375–412. [495, 497]
E., Kaltofen (1990), Polynomial factorization 1982–1986. In Computers in Mathematics, eds. D. V., Chudnovsky and R. D., Jenks, Marcel Dekker, Inc., New York, 285–309. [419]
E., Kaltofen (1992), Polynomial Factorization 1987–1991. In Proceedings of LATIN '92, São Paulo, Brazil, ed. I., Simon. Lecture Notes in Computer Science 583, Springer-Verlag, 294–313. [419]
Erich, Kaltofen (1995a), Effective Noether Irreducibility Forms and Applications. Journal of Computer and System Sciences 50(2), 274–295. [498]
Erich, Kaltofen (1995b), Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Mathematics of Computation 64(210), 777–806. [353]
Erich, Kaltofen (2000), Challenges of Symbolic Computation: My Favourite Open Problems. Journal of Symbolic Computation 29(6), 891–919. With an Additional Open Problem By Robert M. Corless and David J. Jeffrey. [353]
Erich, Kaltofen and Lakshman, Yagati (1988), Improved Sparse Multivariate Polynomial Interpolation Algorithms. In Proceedings of the 1988 International Symposium on Symbolic and Algebraic Computation ISSAC '88, Rome, Italy, ed. P., Gianni. Lecture Notes in Computer Science 358, Springer-Verlag, 467–474. [498]
E., Kaltofen and A., Lobo (1994), Factoring High-Degree Polynomials by the Black Box Berlekamp Algorithm. In Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation ISSAC '94, Oxford, UK, eds. J., von zur Gathen and M., Giesbrecht, ACM Press, 90–98. [404, 405]
Erich, Kaltofen, David R., Musser, and B. David, Saunders (1983), A generalized class of polynomials that are hard to factor. SIAM Journal on Computing 12(3), 473–483. [465]
Erich, Kaltofen and Heinrich, Rolletschek (1989), Computing greatest common divisors and factorizations in quadratic number fields. Mathematics of Computation 53(188), 697–720. [132]
Erich, Kaltofen and B. David, Saunders (1991), On Wiedemann's Method of Solving Sparse Linear Systems. In Algebraic Algorithms and Error-Correcting Codes: AAECC-10, San Juan de Puerto Rico. Lecture Notes in Computer Science 539, Springer-Verlag, 29–38. [340, 351, 404]
Erich, Kaltofen and Victor, Shoup (1997), Fast Polynomial Factorization Over High Algebraic Extensions of Finite Fields. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC '97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 184–188. [420]
Erich, Kaltofen and Victor, Shoup (1998), Subquadratic-Time Factoring of Polynomials over Finite Fields. Mathematics of Computation 67(223), 1179–1197. Extended Abstract in Proceedings of the Twenty-seventh Annual ACM Symposium on the Theory of Computing, Las Vegas NV, ACM Press, 1995, 398–406. [401, 405, 406, 420]
Erich, Kaltofen and Barry M., Trager (1990), Computing with Polynomials Given By Black Boxes for Their Evaluations: Greatest Common Divisors, Factorization, Separation of Numerators and Denominators. Journal of Symbolic Computation 9, 301–320. [496, 498]
Michael, Kaminski, David G., Kirkpatrick, and Nader H., Bshouty (1988), Addition Requirements for Matrix and Transposed Matrix Products. Journal of Algorithms 9, 354–364. [353]
Yasumasa, Kanada (1988), Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of π Calculation. In Supercomputing '88, Volume II: Science and Applications, 117–128. Reprinted in Berggren, Borwein & Borwein (1997), 576–587. [247]
Ravi, Kannan (1987), Algorithmic geometry of numbers. Annual Review of Computer Science 2, 231–267. [496]
A. A., Karatsuba (1995), The Complexity of Computations. Proceedings of the Steklov Institute of Mathematics 211, 169–183. Translated from ТрудьІ Ϻатематичесκого Института имени B. A. Стеклова 211 (1995), 186–202. [247]
A. Карацуьа и Ю., ОФман (1962), Умножение многозначньІх чисел на автоматах. Доκла∂ьι Ака∂емuu Ηаук CCCP 145, 293–294. A. Karatsuba and Yu. Ofman, Multiplication of multidigit numbers on automata, Soviet Physics–Doklady 7 (1963), 595–596. [223, 245]
Alan H., Karp and Peter, Markstein (1997), High-Precision Division and Square Root. ACM Transactions on Mathematical Software 23(4), 561–589. [286]
Richard M., Karp (1972), Reducibility among combinatorial problems. In Complexity of computer computations, eds. Raymond E., Miller and James W., Thatcher, 85–103. Plenum Press, New York. [509, 722]
Michael, Karr (1981), Summation in Finite Terms. Journal of the ACM 28(2), 305–350. [671]
Michael, Karr (1985), Theory of Summation in Finite Terms. Journal of Symbolic Computation 1, 303–315. [671]
Kiran S., Kedlaya and Christopher, Umans (2008), Fast modular composition in any characteristic. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, Philadelphia, PA, IEEE Computer Society Press, 146–155. [751]
Kiran S., Kedlaya and Christopher, Umans (2009), Fast polynomial factorization and modular composition. Merged work of Kedlaya & Umans (2008) and Umans (2008). SIAM Journal on Computing, to appear. Conference version in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, Philadelphia, PA, 481–490. IEEE Computer Society Press. [339, 405, 406, 408, 420]
Walter, Keller-Gehrig (1985), Fast algorithms for the characteristic polynomial. Theoretical Computer Science 36, 309–317. [352]
H., Kempfert (1969), On the Factorization of Polynomials. Journal of Number Theory 1, 116–120. [417, 466]
Thorsten, Kleinjung, Kazumaro, Aoki, Jens, Franke, Arien K., Lenstra, Emmanuel, Thomé, Joppe W., Bos, Pierrick, Gaudry, Alexander Kruppa Peter L., Montgomery, Dag Arne, Osvik, Herman, te Riele, Andrey, Timofeev and Paul, Zimmermann (2010), Factorization of a 768-Bit RSA Modulus. In Advances in Cryptology: Proceedings of CRYPTO '10, Santa Barbara, CA, ed. Tal, Rabin. Lecture Notes in Computer Science 6223, Springer-Verlag, Berlin, Heidelberg, New York, 333–350. [542]
Arnold, Knopfmacher (1995), Enumerating basic properties of polynomials over a finite field. South African Journal of Science 91, 10–11. [419]
Arnold, Knopfmacher and John, Knopfmacher (1993), Counting irreducible factors of polynomials over a finite field. Discrete Mathematics 112, 103–118. [419]
Arnold, Knopfmacher and Richard, Warlimont (1995), Distinct degree factorizations for polynomials over a finite field. Transactions of the American Mathematical Society 347(6), 2235–2243. [419]
Donald E., Knuth (1970), The analysis of algorithms. In Proceedings of the International Congress of Mathematicians 1970, Nice, France, vol. 3, 269–274. [332, 724]
Donald E., Knuth (1993), Johann Faulhaber and sums of powers. Mathematics of Computation 61(203), 277–294. [670]
Donald E., Knuth (1997), The Art of Computer Programming, vol. 1, Fundamental Algorithms. Addison-Wesley, Reading MA, 3rd edition. First edition 1969. [308]
Donald E., Knuth (1998), The Art of Computer Programming, vol. 2, Seminumerical Algorithms. Addison-Wesley, Reading MA, 3rd edition. First edition 1969. [25, 40, 61, 62, 88, 90, 247, 286, 417, 505, 531, 567]
Donald E., Knuth and Luis Trabb, Pardo (1976), Analysis of a simple factorization algorithm. Theoretical Computer Science 3(3), 321–348. [567]
Neal, Koblitz (1987a), A Course in Number Theory and Cryptography. Graduate Texts in Mathematics 114, Springer-Verlag, New York. [531, 568]
Neal, Koblitz (1987b), Elliptic Curve Cryptosystems. Mathematics of Computation 48(177), 203–209. [580]
Helge, von Koch (1904), Sur une courbe continue sans tangente obtenue par une construction géométrique élémentaire. Arkiv för matematik, astronomi och fysik 1, 681–702. [287]
Wolfram, Koepf (1995), Algorithms for m-fold Hypergeometric Summation. Journal of Symbolic Computation 20, 399–417. [671]
Wolfram, Koepf (1998), Hypergeometric Summation. Advanced Lectures in Mathematics, Friedrich Vieweg & Sohn, Braunschweig / Wiesbaden. [670, 697]
János, Kollár (1988), Sharp effective Nullstellensatz. Journal of the American Mathematical Society 1(4), 963–975. [618]
Alwin, Korselt (1899), Problème chinois. L'Intermédiaire des Mathématiciens 6, p. 143. [532]
Henrik, Koy and Claus Peter, Schnorr (2001a), Segment LLL-Reduction of Lattice Bases. In Cryptography and Lattices, International Conference (CaLC 2001), Providence RI, ed. Joseph H., Silverman. Lecture Notes in Computer Science 2146, Springer-Verlag, 67–80. [497]
Henrik, Koy and Claus Peter, Schnorr (2001b), Segment LLL-Reduction with Floating Point Orthogonalization. In Cryptography and Lattices, International Conference (CaLC 2001), Providence RI, ed. Joseph H., Silverman. Lecture Notes in Computer Science 2146, Springer-Verlag, 81–96. [497]
Dexter, Kozen and Susan, Landau (1986), Polynomial Decomposition Algorithms. Technical Report 86-773, Department of Computer Science, Cornell University, Ithaca NY. [752]
Dexter, Kozen and Susan, Landau (1989), Polynomial Decomposition Algorithms. Journal of Symbolic Computation 7, 445–456. An earlier version was published as Kozen & Landau (1986). [576, 581]
Leon G., Kraft Jr. (1949), A Device for Quantizing, Grouping, and Coding Amplitude Modulated Pulses. M.Sc. thesis, Electrical Engineering Department, M.I.T. [307]
M., Kraïtchik (1926), Théorie des Nombres, vol. II. Gauthier-Villars, Paris. [567, 727, 728]
J., Krajíček (1995), Bounded arithmetic, propositional logic and complexity theory. Encyclopedia of Mathematics and its Applications 60, Cambridge University Press, Cambridge, UK. [697]
L., Kronecker (1873), Die verschiedenen Sturmschen Reihen und ihre gegenseitigen Beziehungen. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, 117–154. [197]
L., Kronecker (1878), Über Sturmsche Functionen. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, 95–121. Werke, Zweiter Band, ed. K., Hensel, Leipzig, 1897, 37–70. Reprint by Chelsea Publishing Co., New York, 1968. [197]
L., Kronecker (1881a), Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, 535–600. Werke, Zweiter Band, ed. K., Hensel, Leipzig, 1897, 113–192. Reprint by Chelsea Publishing Co., New York, 1968. [132, 137, 353]
L., Kronecker (1881b), Auszug aus einem Briefe des Herrn Kronecker an E. Schering. Nachrichten der Akademie der Wissenschaften, Göttingen, 271–279. [197]
L., Kronecker (1882), Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Journal für die reine und angewandte Mathematik 92, 1–122. Werke, Zweiter Band, ed. K. Hensel, Leipzig, 1897, 237–387. Reprint by Chelsea Publishing Co., New York, 1968. [247, 465]
Leopold, Kronecker (1883), Die Zerlegung der ganzen Grössen eines natürlichen Rationalitäts-Bereichs in ihre irreductibeln Factoren. Journal für die reine und angewandte Mathematik 94, 344–348. Werke, Zweiter Band, ed. K. HENSEL, Leipzig, 1897, 409–416. Reprint by Chelsea Publishing Co., New York, 1968. [465]
A. H. ΚрьΙлов [A. N., Krylov] (1931), О численном решении уравнения, которьІм в технических воπросах определяются частотьІ мальІх колебаний материальньІх систем (On numerical solutions which determine the frequencies of small oscillations of material systems in technical problems). ИзՅесмuя Ака∂емuu Наук CCCP, Ом∂епенuе Мамемамuческuх u есмесмՅенньιх наук (Bulletin de l'académie des sciences de l'URSS, Classe des sciences mathématiques et naturelles) 4, 491–539. [353]
Y. H., Ku and Xiaoguang, Sun (1992), The Chinese Remainder Theorem. Journal of the Franklin Institute 329, 93–97. [131]
Klaus, Kühnle and Ernst W., Mayr (1996), Exponential Space Computation of Gröbner Bases. In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation ISSAC '96, Zürich, Switzerland, ed. Lakshman, Y. N., ACM Press, 63–71. [616, 617]
H. T., Kung (1974), On Computing Reciprocals of Power Series. Numerische Mathematik 22, 341–348. [286]
J. C., Lafon (1983), Summation in Finite Terms. In Computer Algebra, Symbolic and Algebraic Computation, eds. B., Buchberger, G. E., Collins, and R., Loos, 71–77. Springer-Verlag, New York, 2nd edition. [671]
J. C., Lagarias (1982a), Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators. Transactions of the American Mathematical Society 272(2), 545–554. [509]
J. C., Lagarias (1982b), Best simultaneous Diophantine approximations. II. Behavior of consecutive best approximations. Pacific Journal of Mathematics 102(1), 61–88. [509]
J. C., Lagarias (1985), The computational complexity of simultaneous Diophantine approximation problems. SIAM Journal on Computing 14(1), 196–209. [506, 509]
J. C., Lagarias (1990), Pseudorandom Number Generators in Cryptography and Number Theory. In Cryptology and Computational Number Theory, ed. Carl, Pomerance. Proceedings of Symposia in Applied Mathematics 42, American Mathematical Society, 115–143. [509, 580]
J. C., Lagarias and A. M., Odlyzko (1977), Effective Versions of the Chebotarev Density Theorem. In Algebraic Number Fields, ed. A., Fröhlich, 409–464. Academic Press, London. [443]
J. C., Lagarias and A. M., Odlyzko (1985), Solving Low-Density Subset Sum Problems. Journal of the ACM 32(1), 229–246. [509]
Joseph Louis, de Lagrange (1759), Recherches sur la méthode de maximis et minimis. Miscellanea Taurinensia 1. Œuvres, publiées par J.-A. Serret, vol. 1, 1867, Gauthier-Villars, Paris, 1–20. [131]
Joseph Louis, de Lagrange (1769), Sur la résolution des équations numériques. Mémoires de l'Académie des Sciences et Belles-Lettres de Berlin 23. Œuvres, publiées par J.-A. Serret, vol. 2, 1868, Gauthier-Villars, Paris, 539–578. [419]
Joseph Louis, de Lagrange (1770a), Additions au mémoire sur la résolution des équations numériques. Mémoires de l'Académie des Sciences et Belles-Lettres de Berlin 24. Œuvres, publiées par J.-A. Serret, vol. 2, 1868, Gauthier-Villars, Paris, 581–652. [90]
Joseph Louis, de Lagrange (1770b), Nouvelle méthode pour résoudre les problèmes indéterminés en nombres entiers. Mémoires de l'Académie des Sciences et Belles-Lettres de Berlin 24. Œuvres, publiées par J.-A. Serret, vol. 2, 1868, Gauthier-Villars, Paris, 655–726. [131]
Joseph Louis, de Lagrange (1795), Sur l'usage des courbes dans la solution des Problèmes. In Leçons élémentaires sur les mathématiques, Leçon cinquième. École Polytechnique, Paris. Œuvres, publiées par J.-A. Serret, vol. 7, 1877, Gauthier-Villars, Paris, 271–287. [131, 728]
Joseph Louis, de Lagrange (1798), Additions aux éléments d'algèbre d'Euler. Analyse indéterminée. In Leonhard, Euler, Éléments d'algèbre, St. Petersburg. Œuvres, publiées par J.-A. Serret, vol. 7, 1877, Gauthier-Villars, Paris, 5–180. [90, 91]
Lakshman, Y. N. (1990), On the Complexity of Computing a Gröbner Basis for the Radical of a Zero Dimensional Ideal. In Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, Baltimore MD, ACM Press, 555–563. [618]
B. A., LaMacchia And A. M., Odlyzko (1990), Solving large sparse linear systems over finite fields. In Advances in Cryptology: Proceedings of CRYPTO '90, Santa Barbara, CA. Lecture Notes in Computer Science 537, Springer-Verlag, Berlin and New York, 109–133. [353]
Larry A., Lambe, ed. (1997), Special Issue on Applications of Symbolic Computation to Research and Education. Journal of Symbolic Computation 23(5/6). [21]
Lambert, (1761), Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques. Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin 17, 265–322. Reprint of pages 265–276 in Berggren, Borwein & Borwein (1997), 129–140. [82]
Gabriel, Lamé (1844), Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers. Comptes Rendus de l'Académie des Sciences Paris 19, 867–870. [61]
C., Lanczos (1952), Solutions of systems of linear equations by minimized iterations. Journal of Research of the National Bureau of Standards 49, 33–53. [353]
E., Landau (1905), Sur quelques théorèmes de M. Petrovitch relatifs aux zéros des fonctions analytiques. Bulletin de la Société Mathématique de France 33, 251–261. [165]
F., Landry (1880), Note sur la décomposition du nombre 264 + 1 (Extrait). Comptes Rendus de l'Académie des Sciences Paris 91, p. 138. [542]
Serge, Lang (1983), Fundamentals of Diophantine Geometry. Springer-Verlag, New York. [498]
Tanja, Lange and Arne, Winterhof (2000), Factoring polynomials over arbitrary finite fields. Theoretical Computer Science 234, 301–308. [421]
de la Place, (1772), Recherches sur le calcul intégral et sur le système du monde. Mémoires de l'Académie Royale des Sciences II. Œuvres complètes de Laplace, vol. 8, Gauthier-Villars, Paris, 1891, 367–501. [724]
Daniel, Lauer (2000), Effiziente Algorithmen zur Berechnung von Resultanten und Subresultanten. Berichte aus der Informatik, Shaker Verlag, Aachen. PhD thesis, University of Bonn, Germany. [198, 466]
D., Lazard and R., Rioboo (1990), Integration of Rational Functions: Rational Computation of the Logarithmic Part. Journal of Symbolic Computation 9, 113–115. [640]
V.-A., Lebesgue (1847), Sur le symbole a et quelques-unes de ses applications. Journal de Mathématiques Pures et Appliquées 12, 497–517. [533]
A. M., Legendre (1785), Recherches d'analyse indéterminée. Mémoires de l'Académie Royale des Sciences, 465–559. [198, 418, 420, 466, 468, 569]
A. M., le Gendre (1798, An VI), Essai sur la théorie des nombres. Duprat, Paris. [418, 533, 728]
D. J., Lehmann (1982), On primality tests. SIAM Journal on Computing 11, 374–375. [537]
D. H., Lehmer (1930), An extended theory of Lucas' functions. Annals of Mathematics, Series II 31, 419–448. [530]
D. H., Lehmer (1935), On Lucas's test for the primality of Mersenne's numbers. Journal of the London Mathematical Society 10, 162–165. [530]
D. H., Lehmer (1938), Euclid's algorithm for large numbers. The American Mathematical Monthly 45, 227–233. [332]
D. H., Lehmer and R. E., Powers (1931), On factoring large numbers. Bulletin of the American Mathematical Society 37, 770–776. [569]
Gottfried Wilhelm, Leibniz (1683), Draft letter to Tschirnhaus. In Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern, Erster Band, ed. C. I., Gerhardt, 446–450. Mayer & Müller, Berlin, 1899. Reprinted by Georg Olms Verlag, Hildesheim, 1987. [197]
Gottfried Wilhelm, Leibniz (1697), Nova algebrae promotio. Undated manuscript, c. 1697. In Mathematische Schriften, vol. 7, ed. C. I., Gerhardt, 154–189. Halle, 1863. In: Gesammelte Werke aus den Handschriften der Königlichen Bibliothek zu Hannover, Band VII, Kapitel XV, reprinted by Georg Olms Verlag, Hildesheim, 1971. [88]
Gottfried Wilhelm, Leibniz (1701), Initia mathematica. De ratione et proportione. Undated manuscript, c. 1701. In Mathematische Schriften, vol. 7, ed. C. I., Gerhardt, 1863, 40–49. Reprinted by Georg Olms Verlag, Hildesheim, 1971. [89]
Gothofredus Wilhelmus, Leibnitz [Gottfried Wilhelm Leibniz] (1703), Continuatio analyseos quadraturarum rationalium. Acta eruditorum, 19–26. [640]
Franz, Lemmermeyer (1995), The Euclidean algorithm in algebraic number fields. Expositiones Mathematicae 13, 385–416. [724]
Arjen K., Lenstra (1984), Factoring Polynomials over Algebraic Number Fields. In Proceedings of the 11th International Symposium Mathematical Foundations of Computer Science 1984, Praha, Czechoslovakia. Lecture Notes in Computer Science 176, 389–396. [465]
Arjen K., Lenstra (1987), Factoring multivariate polynomials over algebraic number fields. SIAM Journal on Computing 16, 591–598. [465]
Arjen K., Lenstra (1990), Primality Testing. In Cryptology and Computational Number Theory, ed. Carl, Pomerance. Proceedings of Symposia in Applied Mathematics 42, American Mathematical Society, 13–25. [531]
Arjen K., Lenstra and Hendrik W., Lenstra Jr. (1990), Algorithms in Number Theory. In Handbook of Theoretical Computer Science, vol. A, ed. J., van Leeuwen, 673–715. Elsevier Science Publishers B.V., Amsterdam, and The MIT Press, Cambridge MA. [531]
Arjen K., Lenstra and Hendrik W., Lenstra Jr., eds. (1993), The development of the number field sieve. Lecture Notes in Mathematics 1554, Springer-Verlag, Berlin. [569]
A. K., Lenstra, H. W., Lenstra Jr., and L., Lovász (1982), Factoring Polynomials with Rational Coefficients. Mathematische Annalen 261, 515–534. [474, 497, 506]
Arjen K., Lenstra, Hendrik W., Lenstra Jr., M. S., Manasse, and J. M., Pollard (1990), The number field sieve. In Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, Baltimore MD, ACM Press, 564–572. [569]
Arjen K., Lenstra, Hendrik W., Lenstra Jr., M. S., Manasse, and J. M., Pollard (1993), The factorization of the ninth Fermat number. Mathematics of Computation 61(203), 319–349. [534, 542]
A. K., Lenstra and M. S., Manasse (1990), Factoring by electronic mail. In Advances in Cryptology: Proceedings of EUROCRYPT 1989, Houthalen, Belgium. Lecture Notes in Computer Science 434, Springer-Verlag, Berlin, 355–371. [531]
Hendrik W., Lenstra Jr. (1979a), Euclidean Number Fields 1. The Mathematical Intelligencer 2(1), 6–15. [724]
Hendrik W., Lenstra Jr. (1979b), Miller's primality test. Information Processing Letters 8(2), 86–88. [532]
Hendrik W., Lenstra Jr. (1980a), Euclidean Number Fields 2. The Mathematical Intelligencer 2(2), 73–77. [724]
Hendrik W., Lenstra Jr. (1980b), Euclidean Number Fields 3. The Mathematical Intelligencer 2(2), 99–103. [724]
H. W., Lenstra Jr. (1987), Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673. [541, 557, 558, 565, 568]
H. W., Lenstra Jr. (1990), Algorithms for finite fields. In Number theory and cryptography, ed. J. H., Loxton, London Mathematical Society Lecture Note Series 154, 76–85. Cambridge University Press, Cambridge, UK. [569]
H. W., Lenstra Jr. (1991), Finding isomorphisms between finite fields. Mathematics of Computation 56(193), 329–347. [419]
H. W., Lenstra Jr. and Carl, Pomerance (1992), A rigorous time bound for factoring integers. Journal of the American Mathematical Society 5(3), 483–516. [569]
A. H. M., Levelt (1997), The cycloheptane molecule – a challenge to computer algebra. Invited lecture given at the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC '97, Maui HI. [698]
L. A., Levin (1973), Universal sequential search problems. Problems of Information Transmission 9, 265–266. Translated from Problemy Peredachi Informatsii 9(3) (1973), 115–116. [724]
Daniel, Lewin and Salil, Vadhan (1998), Checking Polynomial Identities over any Field: Towards a Derandomization? In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, Dallas TX, ACM Press, 438–447. [199]
T., Lickteig (1987), The computational complexity of division in quadratic extension fields. SIAM Journal on Computing 16, 278–311.
Thomas, Lickteig and Marie-Françoise, Roy (1996), Cauchy Index Computation. Calcolo 33, 331–357. [184, 199, 332]
Thomas, Lickteig and Marie-Françoise, Roy (2001), Sylvester-Habicht Sequences and Fast Cauchy Index Computation. Journal of Symbolic Computation 31, 315–341. [184, 199, 332]
Rudolf, Lidl and Harald, Niederreiter (1997), Finite Fields. Encyclopedia of Mathematics and its Applications 20, Cambridge University Press, Cambridge, UK, 2nd edition. First published by Addison-Wesley, Reading MA, 1983. [421, 711]
F., Lindemann (1882), Über die Zahl π. Mathematische Annalen 20, 213–225. [82]
J. H., van Lint (1982), Introduction to Coding Theory. Graduate Texts in Mathematics 86, Springer-Verlag, New York. [215]
Joseph, Liouville (1833a), Sur la détermination des Intégrales dont la valeur est algébrique. Journal de l'École Polytechnique 14, Premier Mémoire: 124–148, Second Mémoire: 149–193. [640]
Joseph, Liouville (1833b), Note sur la détermination des intégrales dont la valeur est algébrique. Journal für die reine und angewandte Mathematik 10, 347–359. Errata 11 (1834), 406. [640]
Joseph, Liouville (1835), Mémoire sur l'intégration d'une classe de fonctions transcendantes. Journal für die reine und angewandte Mathematik 13(2), 93–118. [623, 640]
John D., Lipson (1971), Chinese remainder and interpolation algorithms. In Proceedings 2nd ACM Symposium on Symbolic and Algebraic Manipulation, Los Angeles CA, ed. S. R., Petrick, ACM Press, 372–391. [306]
John D., Lipson (1981), Elements of Algebra and Algebraic Computing. Addison-Wesley, Reading MA. [247]
Petr, Lisoněk, Peter, Paule, and Volker, Strehl (1993), Improvement of the degree setting in Gosper's algorithm. Journal of Symbolic Computation 16, 243–258. [670, 671]
Daniel B., Lloyd (1964), Factorization of the general polynomial by means of its homomorphic congruential functions. The American Mathematical Monthly 71, 863–870. [419]
Daniel B., Lloyd and Harry, Remmers (1966), Polynomial factor tables over finite fields. Mathematical Algorithms 1, 85–99. [419]
Henri, Lombardi, Marie-Françoise, Roy, and Mohab Safey El, Din (2000), New Structure Theorem for Subresultants. Journal of Symbolic Computation 29, 663–689. [199]
Rüdiger, Loos (1983), Computing rational zeroes of integral polynomials by p-adic expansion. SIAM Journal on Computing 12(2), 286–293. [466]
S. C., Lu and L. N., Lee (1979), A simple and effective public-key cryptosystem. COMSAT Technical Review 9(1), 15–24. [509]
Édouard Lucas (1878), Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics 1, I: 184–240, II: 289–321. [530]
Paul, Luckey (1951), Die Rechenkunst bei Ǧamšīd b. Masͨūd al-Kāšī. Abhandlungen für die Kunde des Morgenlandes, XXXI, 1, Kommissionsverlag Franz Steiner GmbH, Wiesbaden. Herausgegeben von der Deutschen Morgenländischen Gesellschaft. [725]
P., Luckey (1953), Der Lehrbrief über den Kreisumfang (Ar-risāla al-muḥītīya) von Ǧamšīd B. Masḥūd Al-Kāšī. Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin, Klasse für Mathematik und allgemeine Naturwissenschaften 6, Akademie-Verlag, Berlin. [90]
J., van de Lune, H. J. J., te Riele, and D. T., Winter (1986), On the Zeros of the Riemann Zeta Function in the Critical Strip. IV. Mathematics of Computation 46(174), 667–681. [533]
Keju, Ma and Joachim, von zur Gathen (1990), Analysis of Euclidean Algorithms for Polynomials over Finite Fields. Journal of Symbolic Computation 9, 429–455. [62]
F. S., Macaulay (1902), Some formulæ in elimination. Proceedings of the London Mathematical Society 35, 3–27. [197, 619]
F. S., Macaulay (1916), The algebraic theory of modular systems. Cambridge University Press, Cambridge, UK. Reissued 1994. [197, 619, 728, 729]
F. S., Macaulay (1922), Note on the resultant of a number of polynomials of the same degree. Proceedings of the London Mathematical Society, Second Series 21, 14–21. [197, 619]
D., Mack (1975), On rational integration. Technical Report UCP-38, Department of Computer Science, University of Utah. [642]
Colin, Maclaurin (1742), A treatise of fluxions. 2 volumes, Edinburgh. 2nd ed., London, 1801; French translation Paris, 1749. [286]
F. J., MacWilliams and N. J. A., Sloane (1977), The Theory of Error-Correcting Codes. Mathematical Library 16, North-Holland, Amsterdam. [215]
Dietrich, Mahnke (1912/1913), Leibniz auf der Suche nach einer allgemeinen Primzahlgleichung. Bibliotheca Mathematica, Serie 3, 13, 29–61. [88, 531]
Yiu-Kwong, Man (1993), On Computing Closed Forms for Indefinite Summations. Journal of Symbolic Computation 16, 355–376. [671]
Benoît B., Mandelbrot (1977), The fractral geometry of nature. Freeman. [278]
Ю. И., Манин (1956), О сравнениях третьей степени по простому модулю. Извесмuя Ака∂емuu Ηаук CCCP, Серuя Маmемаuческая 20, 673–678. YU. I. MANIN, On cubic congruences to a prime modulus, American Mathematical Society Translations, Series 2, 13 (1960), 1–7. [568]
J. L., Massey (1965), Step by step decoding of the Bose-Chaudhuri-Hocquenghem codes. IEEE Transactions on Information Theory IT–11, 580–585. [215]
Ю. B., Матиясевич (1970), Диоϕантовость перечислимих множеств. Докла∂ьι Ακа∂емuu Наук CCCP 191(2), 279–282. Yu. V. Matiyasevich, Enumerable sets are Diophantine, Soviet Mathematics Doklady 11(2), 354–358. [89]
Юрий B., МатиясEвич (1993), Докла∂ьι Ακа∂емuu Наук. Nauka, Moscow. Yuri V. Matiyasevich, Hilbert's Tenth Problem, Foundations of Computing Series, The MIT Press, Cambridge MA, 1993. [89, 640]
Ueli M., Maurer and Stefan, Wolf (1999), The relationship between breaking the Dif?e-Hellman protocol and computing discrete logarithms. SIAM Journal on Computing 28(5), 1689–1721. [580]
Ernst W., Mayr (1984), An algorithm for the general Petri net reachability problem. SIAM Journal on Computing 13(3), 441–460. [697]
Ernst, Mayr (1989), Membership in Polynomial Ideals over Q Is Exponential Space Complete. In Proceedings of the 6th Annual Symposium on Theoretical Aspects of Computer Science STACS '89, Paderborn, Germany, eds. B., Monien and R., Cori. Lecture Notes in Computer Science 349, Springer-Verlag, 400–406. [616]
Ernst W., Mayr (1992), Polynomial ideals and applications. Mitteilungen der Mathematischen Gesellschaft in Hamburg 12(4), 1207–1215. Festschrift zum 300jährigen Bestehen der Gesellschaft. [616, 697]
Ernst W., Mayr (1995), On Polynomial Ideals, Their Complexity, and Applications. In Proceedings of the 10th International Conference on Fundamentals of Computation Theory FCT '95, Dresden, Germany, ed. Horst, Reichel. Lecture Notes in Computer Science 965, Springer-Verlag, 89–105. [616, 697]
Ernst W., Mayr (1997), Some complexity results for polynomial ideals. Journal of Complexity 13, 303–325. [618]
Ernst W., Mayr and Albert R., Meyer (1982), The Complexity of the Word Problems for Commutative Semigroups and Polynomial Ideals. Advances in Mathematics 46, 305–329. [616, 617, 618]
Ernst W., Mayr and Stephan, Ritscher (2010), Degree Bounds for Gröbner Bases of Low-Dimensional Polynomial Ideals. Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation ISSAC2010, Munich, Germany, 21–27. [617]
Kevin S., McCurley (1990), The Discrete Logarithm Problem. In Cryptology and Computational Number Theory, ed. Carl, Pomerance. Proceedings of Symposia in Applied Mathematics 42, American Mathematical Society, 49–74. [580]
Robert J., McEliece (1969), Factorization of Polynomials over Finite Fields. Mathematics of Computation 23, 861–867. [419]
Alfred, Menezes (1993), Elliptic curve public key cryptosystems. Kluwer Academic Publishers, Boston MA. [580]
Ralph C., Merkle and Martin E., Hellman (1978), Hiding information and signatures in trapdoor knapsacks. IEEE Transactions on Information Theory IT–24(5), 525–530. [503, 504, 509, 576]
Marin, Mersenne (1636), Harmonie universelle contenant la théorie et la pratique de la musique. Sebastien Cramoisy, Paris. Reprinted by Centre National de la Recherche Scientifique, Paris, 1975. [86]
F., Mertens (1897), Über eine zahlentheoretische Function. Sitzungsberichte der Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Classe 106, 761–830. [508]
Nicholas, Metropolis and S., Ulam (1949), The Monte Carlo Method. Journal of the American Statistical Association 44, 335–341. [198]
Shawna Meyer, Eikenberry and Jonathan P., Sorenson (1998), Efficient algorithms for computing the Jacobi symbol. Journal of Symbolic Computation 26(4), 509–523. [533]
M., Mignotte (1974), An Inequality About Factors of Polynomials. Mathematics of Computation 28(128), 1153–1157. [198]
M., Mignotte (1982), Some Useful Bounds. In Computer Algebra, Symbolic and Algebraic Computation, eds. B., Buchberger, G. E., Collins, and R., Loos, 259–263. Springer-Verlag, New York, 2nd edition. [198]
Maurice, Mignotte (1988), An Inequality about Irreducible Factors of Integer Polynomials. Journal of Number Theory 30, 156–166. [198]
Maurice, Mignotte (1989), Mathématiques pour le calcul formel. Presses Universitaires de France, Paris. English translation: Mathematics for Computer Algebra, Springer-Verlag, New York, 1992. [198]
Maurice, Mignotte and Philippe, Glesser (1994), On the Smallest Divisor of a Polynomial. Journal of Symbolic Computation 17, 277–282. [198]
Maurice, Mignotte and C., Schnorr (1988), Calcul des racines d-ièmes dans un corps fini. Comptes Rendus de l'Académie des Sciences Paris 290, 205–206. [421]
Ш. Е., Микеладзе [Sh. E., Mikeladze] (1948), О разложении определителя, элементами которого служат полиномьІ (On the expansion of a determinant whose entries are polynomials). Πрuкла∂ная маmемаmuка u механuка (Prikladnaya matematika i mekhanika) 12, 219–222. [132]
Gary L., Miller (1976), Riemann's Hypothesis and Tests for Primality. Journal of Computer and System Sciences 13, 300–317. [532]
Victor S., Miller (1986), Use of Elliptic Curves in Cryptography. In Advances in Cryptology: Proceedings of CRYPTO '85, Santa Barbara, CA, ed. Hugh C., Williams. Lecture Notes in Computer Science 218, Springer-Verlag, Berlin, 417–426. [580]
H., Minkowski (1910), Geometrie der Zahlen. B. G. Teubner, Leipzig. [496]
R. T., Moenck (1973), Fast computation of gcd's. In Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, Austin TX, ACM Press, 142–151. [332]
Robert T., Moenck (1976), Practical Fast Polynomial Multiplication. In Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation SYMSAC '76, Yorktown Heights NY, ed. R. D., Jenks, ACM Press, 136–148. [247]
Robert T., Moenck (1977a), On the Efficiency of Algorithms for Polynomial Factoring. Mathematics of Computation 31(137), 235–250. [421]
Robert, Moenck (1977b), On computing closed forms for summation. In Proceedings of the 1977 MACSYMA Users Conference, Berkeley CA, NASA, Washington DC, 225–236. [671, 673]
R., Moenck and A., Borodin (1972), Fast modular transform via division. In Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, Yorktown Heights NY, IEEE Press, New York, 90–96. [306]
Michael, Moeller (1999), Good non-zeros of polynomials. ACM SIGSAM Bulletin 33(3), 10–11. [199]
H. Michael, Möller and Ferdinando, Mora (1984), Upper and lower bounds for the degree of Gröbner bases. In Proceedings of EUROSAM '84, Cambridge, UK, ed. John, Fitch. Lecture Notes in Computer Science 174, Springer-Verlag, New York, 172–183. [618]
Louis, Monier (1980), Evaluation and comparison of two ef?cient probabilistic primality testing algorithms. Theoretical Computer Science 12, 97–108. [532, 533]
Peter L., Montgomery (1985), Modular Multiplication Without Trial Division. Mathematics of Computation 44(170), 519–521. [288]
Peter L., Montgomery (1991), Factorization of X216091 + X +1 mod 2—A problem of Herb Doughty. Manuscript. [280]
Peter Lawrence, Montgomery (1992), An FFT Extension of the Elliptic Curve Method of Factorization. PhD thesis, University of California, Los Angeles CA. http://research.microsoft.com/en-us/um/people/petmon/thesis.pdf. [287, 308]
Peter L., Montgomery (1995), A Block Lanczos Algorithm for Finding Dependencies over GF(2). In Advances in Cryptology: Proceedings of EUROCRYPT1995, Saint-Malo, France, eds. Louis C., Guillou and Jean-Jacques, Quisquater. Lecture Notes in Computer Science 921, Springer-Verlag, 106–120. [353]
Eliakim Hastings, Moore (1896), A doubly-infinite system of simple groups. In Mathematical papers read at the International Mathematical Congress: held in connection with the World's Columbian exposition, Chicago, 1893, Macmillan, New York, 208–242. [88]
Robert Edouard, Moritz (1914), Memorabilia Mathematica. The Mathematical Association of America. [729]
Michael A., Morrison and John, Brillhart (1971), The factorization of F7. Bulletin of the American Mathematical Society 77(2), p. 264. [542, 568]
Michael A., Morrison and John, Brillhart (1975), A Method of Factoring and the Factorization of F7. Mathematics of Computation 29(129), 183–205. [541, 568]
Joel, Moses and David Y. Y., Yun (1973), The EZGCD Algorithm. In Proceedings of the ACM National Conference, Atlanta GA, 159–166. [198, 466]
Rajeev, Motwani and Prabhakar, Raghavan (1995), Randomized Algorithms. Cambridge University Press, Cambridge, UK. [88, 198]
Thom, Mulders (1997), A note on subresultants and the Lazard/Rioboo/Trager formula in rational function integration. Journal of Symbolic Computation 24(1), 45–50. [199, 640]
T., Mulders and A., Storjohann (2000), On Lattice Reduction for Polynomial Matrices. Technical Report 356, Department of Computer Science, ETH Zürich. 26 pages, ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/3xx/356.ps.gz. [501]
R. C., Mullin, I. M., Onyszchuk, S. A., Vanstone, and R. M., Wilson (1989), Optimal normal bases in GF(pn). Discrete Applied Mathematics 22, 149–161. [88]
David R., Musser (1971), Algorithms for Polynomial Factorization. PhD thesis, Computer Science Department, University of Wisconsin. Technical Report #134, 174 pages. [465]
Mats, Näslund (1998), Bit Extraction, Hard-Core Predicates, and the Bit Security of RSA. PhD thesis, Department of Numerical Analysis and Computing Science, Kungl Tekniska Högskolan (Royal Institute of Technology), Stockholm. [580]
Isaac, Newton (1691/1692), De quadratura curvarum. The revised and augmented treatise. Unpublished manuscript. In: Derek T., Whiteside, The mathematical papers of Isaac Newton vol. VII, Cambridge University Press, Cambridge, UK, 1976, pp. 48–128. [641]
Isaac, Newton (1707), Arithmetica Universalis, sive de compositione et resolutione arithmetica liber. J. Senex, London. English translation as Universal Arithmetick: or, A Treatise on Arithmetical composition and Resolution, translated by the late Mr. Raphson and revised and corrected by Mr. Cunn, London, 1728. Reprinted in: Derek T. Whiteside, The mathematical works of Isaac Newton, Johnson Reprint Co, New York, 1967, p. 4 ff. [61, 203, 725, 726]
Isaac, Newton (1710), Quadrature of Curves. In Lexicon Technicum. Or, an Universal Dictionary of Arts and Sciences, vol. 2, John Harris. Reprinted in: Derek T. Whiteside, The mathematical works of Isaac Newton, vol. 1, Johnson Reprint Co, New York, 1967. [286]
Phong Q., Nguyen and Jacques, Stern (2001), The Two Faces of Lattices in Cryptology. In Cryptography and Lattices, International Conference (CaLC 2001), Providence RI, ed. Joseph H., Silverman. Lecture Notes in Computer Science 2146, Springer-Verlag, 146–180. [509, 580]
Thomas R., Nicely (1996), Enumeration to 1014 of the Twin Primes and Brun's Constant. Virginia Journal of Science 46(3), 195–204. [83]
H., Niederreiter (1986), Knapsack-type cryptosystems and algebraic coding theory. Problems of Control and Information Theory 15, 159–166. [509]
Harald, Niederreiter (1993a), A New Efficient Factorization Algorithm for Polynomials over Small Finite Fields. Applicable Algebra in Engineering, Communication and Computing 4, 81–87. [420]
H., Niederreiter (1993b), Factorization of Polynomials and Some Linear Algebra Problems over Finite Fields. Linear Algebra and its Applications 192, 301–328. [420]
Harald, Niederreiter (1994a), Factoring polynomials over finite fields using differential equations and normal bases. Mathematics of Computation 62(206), 819–830. [420]
Harald, Niederreiter (1994b), New deterministic factorization algorithms for polynomials over finite fields. In Finite fields: theory, applications and algorithms, eds. G. L., Mullen and P. J.-S., Shiue. Contemporary Mathematics 168, American Mathematical Society, 251–268. [420]
Harald, Niederreiter and Rainer, Göttfert (1993), Factorization of Polynomials over Finite Fields and Characteristic Sequences. Journal of Symbolic Computation 16, 401–412. [420]
Harald, Niederreiter and Rainer, Göttfert (1995), On a new factorization algorithm for polynomials over finite fields. Mathematics of Computation 64(209), 347–353. [420]
Pedro, Nunez (1567), Libro de algebra en arithmetica y geometrica. Iuan Stelfio, widow and heirs, Anvers. [41]
A. M., Odlyzko (1990), The Rise and Fall of Knapsack Cryptosystems. In Cryptology and Computational Number Theory, ed. Carl, Pomerance. Proceedings of Symposia in Applied Mathematics 42, American Mathematical Society, 75–88. [497, 509]
A. M., Odlyzko (1995a), Asymptotic Enumeration Methods. In Handbook of Combinatorics, eds. R., Graham, M., grötschel, and L., Lovász. Elsevier Science Publishers B.V., Amsterdam, and The MIT Press, Cambridge MA. [697]
Andrew M., Odlyzko (1995b), The Future of Integer Factorization. CryptoBytes 1(2), 5–12. [580]
Andrew M., Odlyzko (1995c), Analytic computations in number theory. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, ed. Walter, Gautschi. Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, 451–463. [533]
A. M., Odlyzko and H. J. J., te Riele (1985), Disproof of the Mertens conjecture. Journal für die reine und angewandte Mathematik 357, 138–160. [508]
A. M., Odlyzko and A., Schönhage (1988), Fast algorithms for multiple evaluations of the Riemann zeta function. Transactions of the American Mathematical Society 309(2), 797–809. [533]
Joseph, Oesterlé (1979), Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisée. Société Mathématique de France, Astérisque 61, 165–167. [443]
H., Ong, C. P., Schnorr, and A., Shamir (1984), An efficient signature scheme based on quadratic equations. In Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, Washington DC, ACM Press, 208–216. [509]
Luitzen Johannes, Oosterhoff (1949), Restricted free rotation and cyclic molecules. PhD thesis, Rijksuniversiteit te Leiden. [698]
Alan V., Oppenheim and Ronald W., Schafer (1975), Digital Signal Processing. Prentice-Hall, Inc., Englewood Cliffs NJ. [368]
Alan V., Oppenheim, Alan S., Willsky, and Ian T., Young (1983), Signals and Systems. Prentice-Hall signal processing series, Prentice-Hall, Inc., Englewood Cliffs NJ. [368]
M., Ostrogradsky (1845), De l'intégration des fractions rationnelles. Bulletin de la classe physicomathématique de l'Académie Impériale des Sciences de Saint-Pétersbourg 4(82/83), 145–167. [640]
H., Padé (1892), Sur la représentation approchée d'une fonction par des fractions rationnelles. Annales Scientifiques de l'Ecole Normale Supérieure, 3e série 9, Supplément S3–S93. [132]
Я. П, аһ (1960) О ϲπосбах вцчсления значении многочленов. Усnехu мамемамuческuх Наук 21(1(127)), 103–134. V. Ya. Pan, Methods of computing values of polynomials, Russian Mathematical Surveys 21 (1966), 105-136. [306]
V. Ya., pan (1984), How to multiply matrices faster. Lecture Notes in Computer Science 179, Springer-Verlag, New York. [352]
Victor Y., Pan (1997), Faster Solution of the Key Equation for Decoding BCH Error-Correcting Codes. In Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, El Paso TX, ACM Press, 168–175. [332]
Victor Y., Pan and Xinmao, Wang (2004), On Rational Number Reconstruction and Approximation. SIAM Journal on Computing 33(2), 502–503. [327]
Daniel Nelson Panario, Rodriguez (1997), Combinatorial and Algebraic Aspects of Polynomials over Finite Fields. PhD thesis, Department of Computer Science, University of Toronto. Technical Report 306/97, 154 pages. [419]
Daniel, Panario, Xavier, Gourdon, and Philippe, Flajolet (1998), An Analytic Approach to Smooth Polynomials over Finite Fields. In Algorithmic Number Theory, Third International Symposium, ANTS-HI, Portland, Oregon, USA, ed. J. P., Buhler. Lecture Notes in Computer Science 1423, Springer-Verlag, 226–236. [419]
Daniel, Panario and Bruce, Richmond (1998), Analysis of Ben-Or's Polynomial Irreducibility Test. Random Structures and Algorithms 13(3/4), 439–456. [419, 421]
Daniel, Panario and Alfredo, Viola (1998), Analysis of Rabin's polynomial irreducibility test. In Proceedings of LATIN ′98, Campinas, Brazil, eds. Claudio L., Lucchesi and Arnaldo V., Moura. Lecture Notes in Computer Science 1380, Springer-Verlag, 1–10. [419, 421]
Christos H., Papadimitriou (1993), Computational complexity. Addison-Wesley, Reading MA. [721]
David, Parsons and John, Canny (1994), Geometric Problems in Molecular Biology and Robotics. In Proceedings 2nd International Conference on Intelligent Systems for Molecular Biology, Palo Alto CA, 322–330. [698]
Peter, Paule (1994), Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type. The Electronic Journal of Combinatorics 1(# R10). 9 pages. [697]
Peter, Paule (1995), Greatest Factorial Factorization and Symbolic Summation. Journal of Symbolic Computation 20, 235–268. [670, 671]
Peter, Paule and Volker, Strehl (1995), Symbolic summation – some recent developments. In Computer Algebra in Science and Engineering, Bielefeld, Germany, August 1994, eds. J., Fleischer, J., Grabmeier, F. W., Hehl, and W., Küchlin, World Scientific, Singapore, 138–162. [671]
Heinz-Otto, Peitgen, Hartmut, Jürgens, and Dietmar, Saupe (1992), Chaos and Fractals: New Frontiersof Sience. Springer-Verlag, New York. [278]
William B., Pennebaker and Joan C., Mitchell (1993), JPEG still image data compression standard. Van Nostrand Reinhold, New York. [368]
Pepin, (1877), Sur la formule 22n + 1. Comptes Rendus des Séances del'Académie des Sciences, Paris 85, 329–331. [530, 538]
Oskar, Perron (1929), Die Lehre von den Kettenbrüchen. B. G., Teubner, Leipzig, 2nd edition. Reprinted by Chelsea Publishing Co., New York. First edition 1913. [90]
James L., Peterson (1981), Petri net theory and the modeling of systems. Prentice-Hall, Inc., Englewood Cliffs NJ. [697]
Marko, Petkovšek (1992), Hypergeometric solutions of linear recurrences with polynomial coefficients. Journal of Symbolic Computation 14, 243–264. [671, 675]
Marko, Petkovšek (1994), A generalization of Gosper's algorithm. Discrete Mathematics 134, 125–131. [671]
Marko, Petkovšek and Bruno, Salvy (1993), Finding All Hypergeometric Solutions of Linear Differential Equations. In Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation ISSAC ′93, Kiev, , ed. Manuel Bronstein, ACM Press, 27–33. [641]
Marko, Petkovšek, Herbert S., Wilf, and Doron, Zeilberger (1996), A=B. A K Peters, Wellesley MA. [697, 729]
Karel, Petr (1937), Über die Reduzibilität eines Polynoms mit ganzzahligen Koeffizienten nach einem Primzahlmodul. Časopis pro pěstování matematiky a fysiky 66, 85–94. [402, 420]
C. A., Petri (1962), Kommunikation mit Automaten. PhD thesis, Universität Bonn. [679]
Eckhard, Pflügel (1997), An Algorithm for Computing Exponential Solutions of First Order Linear Differential Systems. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC ′97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 164–171. [641]
R. G. E., Pinch (1993), Some Primality Testing Algorithms. Notices of the American Mathematical Society 40(9), 1203–1210. [532]
R., Pirastu (1992), Algorithmen zur Summation rationaler Funktionen. Diplomarbeit, Universität Erlangen-Nürnberg, Germany. [670, 673]
Roberto, Pirastu (1996), On Combinatorial Identities: Symbolic Summation and Umbral Calculus. PhD thesis, Johannes Kepler Universität, Linz. [671]
R., Pirastu and V., Strehl (1995), Rational Summation and Gosper-Petkovšek Representation. Journal of Symbolic Computation 20, 617–635. [671]
Toniann, Pitassi (1997), Algebraic Propositional Proof Systems. In Descriptive Complexity and Finite Models: Proceedings of aDIMACS Workshop, January 14-17, 1996, Princeton NJ, eds. Neil, Immerman and Phokion G., Kolaitis. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 31, American Mathematical Society, Providence RI, 215–244. [697]
H. C., Pocklington (1917), The Direct Solution of the Quadratic and Cubic Binomial Congruences with Prime Moduli. Proceedings of the Cambridge Philosophical Society 19, 57–59. [88, 198]
John M., Pollard (1971), The Fast Fourier Transform in a Finite Field. Mathematics of Computation 25(114), 365–374. [247, 280]
John M., Pollard (1974), Theorems on factorization and primality testing. Proceedings of the Cambridge Philosophical Society 76, 521–528. [198, 541, 567]
John M., Pollard (1975), A Monte Carlo method for factorization. BIT 15, 331–334. [198, 541, 545, 568]
C., Pomerance (1982), Analysis and comparison of some integer factoring algorithms. In Computational Methods in Number Theory, Part 1, eds. H. W., Lenstra Jr. and R., Tijdeman, Mathematical Centre Tracts 154, 89–139. Mathematisch Centrum, Amsterdam. [557, 567, 569]
Carl, Pomerance (1985), The quadratic sieve factoring algorithm. In Advancesin Cryptology: Proceedings of EUROCRYPT 1984, Paris, France, eds. T., Beth, N., Cot, and I., Ingemarsson. Lecture Notes in Computer Science 209, Springer-Verlag, Berlin, 169–182. [557]
Carl, Pomerance (1990), Factoring. In Cryptology and Computational Number Theory, ed. Carl, Pomerance. Proceedings of Symposia in Applied Mathematics 42, American Mathematical Society, 27–47. [520, 567]
C., Pomerance, J. L., Selfridge, and S.S., Wagstaff Jr. (1980), The pseudoprimes to 25 · 109. Mathematics of Computation 35, 1003–1025. [532]
Carl, Pomerance and S.S., Wagstaff Jr. (1983), Implementation of the continued fraction integer factoring algorithm. Congressus Numerantium 37, 99–118. [569]
Alfred, van der Poorten (1978), A proof that Euler missed … Apéry's proof of the irrationality of ς(3). The Mathematical Intelligencer 1, 195–203. [697]
Alf, van der Poorten (1996), Notes on Fermat's Last Theorem. Canadian Mathematical Society series of monographs and advanced texts, John Wiley & Sons, New York. [514]
Eugene, Prange (1959), An algorism for factoring Xn – 1 over a finite field. Technical Report AFCRC-TN-59-775, Air Force Cambridge Research Center, Bedford MA. [419, 430]
Paul, Pritchard (1983), Fast Compact Prime Number Sieves (among Others). Journal of Algorithms 4, 332–344. [533]
Paul, Pritchard (1987), Linear prime-number sieves: a family tree. Science of Computer Programming 9, 17–35. [533]
George B., Purdy (1974), A high-security log-in procedure. Communications of the ACM 17(8), 442–445. [581]
Michael O., Rabin (1976), Probabilistic algorithms. In Algorithms and Complexity, ed. J. F., Traub, Academic Press, New York, 21–39. [532]
Michael O., Rabin (1980a), Probabilistic Algorithms for Testing Primality. Journal of Number Theory 12, 128–138. [532]
Michael O., Rabin (1980b), Probabilistic algorithms in finite fields. SIAM Journal on Computing 9(2), 273–280. [421, 424]
Michael O., Rabin (1989), Efficient Dispersal of Information for Security, Load Balancing, and Fault Tolerance. Journal of the Association for Computing Machinery 36(2), 335–348. [131, 215]
J. L., Rabinowitsch (1930), Zum Hilbertschen Nullstellensatz. Mathematische Annalen 102, p. 520. [618]
Bartolomé, Ramos (1482), De musica tractatus. Bologna. [86]
Joseph, Raphson (1690), Analysis Æquationum Universalis seu Ad Æquationes Algebraicas Resolvendas Methodus Generalis, et Expedita, Ex nova Infinitarum serierum Doctrina Deducta ac Demonstrata. Abel Swalle, London. [219]
Alexander A., Razborov (1998), Lower bounds for the polynomial calculus. computational complexity 7(4), 291–324. [697]
Constance, Reid (1970), Hilbert. Springer-Verlag, Heidelberg, 1st edition. Third Printing 1978. [587]
Daniel, Reischert (1995), Schnelle Multiplikation von Polynomen über GF(2) und Anwendungen. Diplomarbeit, Institut für Informatik II, Rheinische Friedrich-Wilhelm-Universität Bonn, Germany. [279]
Daniel, Reischert (1997), Asymptotically Fast Computation of Subresultants. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC ′97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 233–240. [332]
Wolfgang, Reisig (1985), Petri Nets: An Introduction. EATCS Monographs on Theoretical Computer Science 4, Springer-Verlag, Berlin. Translation of the German edition Petrinetze: eine Einführung, Springer-Verlag, 1982. [697]
George W., Reitwiesner (1950), An ENIAC Determination of π and e to more than 2000 Decimal Places. Mathematical Tables and other Aids to Computation 4, 11–15. Reprinted in Berggren, Borwein & Borwein (1997), 277-281. [82]
James, Renegar (1991), Recent Progress on the Complexity of the Decision Problem for the Reals. In Discrete and Computational Geometry: Papers from the DIMACS Special Year, eds. Jacob E., Goodman, Richard, Pollack, and William, Steiger. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 6, American Mathematical Society and ACM, 287–308. [619]
James, Renegar (1992a), On the Computational Complexity of the First-order Theory of the Reals. Part I: Introduction. Preliminaries. The Geometry of Semi-algebraic Sets. The Decision Problem for the Existential Theory of the Reals. Journal of Symbolic Computation 13(3), 255–299. [619]
James, Renegar (1992b), On the Computational Complexity of the First-order Theory of the Reals. Part II: The General Decision Problem. Preliminaries for Quantifier Elimination. Journal of Symbolic Computation 13(3), 301–327. [619]
James, Renegar (1992c), On the Computational Complexity of the First-order Theory of the Reals. Part III: Quantifier Elimination. Journal of Symbolic Computation 13(3), 329–352. [619]
Reynaud, (1824), Traité d'arithmétique à l'usage des élèves qui se destinent à l'école royale polytechnique à l'école spéciale militaire et à l'école de marine. Courcier, Paris, 12th edition. [61]
Daniel, Richardson (1968), Some undecidable problems involving elementary functions of a real variable. Journal ofSymbolicLogic 33(4), 514–520. [640]
Georg, FriedrichBernhard, Riemann (1859), Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, 145–153. Gesammelte Mathematische Werke, ed. Heinrich, Weber, Teubner Verlag, Leipzig, 1892, 177–185. [533]
Robert H., Risch (1969), The problem of integration in finite terms. Transactions of the American Mathematical Society 139, 167–189. [640, 641]
Robert H., Risch (1970), The solution of the problem of integration in finite terms. Bulletin of the American MathematicalSociety 76(3), 605–608. [640, 641]
J. F., Ritt (1948), Integration in Finite Terms. Columbia University Press, New York. [640]
Joseph Fels, Ritt (1950), Differential Algebra. AMS Colloquium Publications XXXIII, American Mathematical Society, Providence RI. Reprint by Dover Publications, Inc., New York, 1966. [619]
R. L., Rivest, A., Shamir, and L. M., Adleman (1978), A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM 21(2), 120–126. [576]
Steven, Roman (1984), The umbral calculus. Pure and applied mathematics 111, Academic Press, Orlando FL. [669]
Lajos, Rónyai (1988), Factoring Polynomials over Finite Fields. Journal of Algorithms 9, 391–400. [421]
Lajos, Rónyai (1989), Galois groups and factoring over finite fields. In Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, Research Triangle Park NC, IEEE Computer Society Press, Los Alamitos CA, 99–104. [421]
Frederic, Rosen (1831), The Algebra of Mohammed ben Musa. Oriental Translation Fund, London. Reprint by Georg Olms Verlag, Hildesheim, 1986. [726]
J. Barkley, Rosser and Lowell, Schoenfeld (1962), Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics 6, 64–94. [527, 532, 536]
Michael, Rothstein (1976), Aspects of symbolic integration and simplification of exponential andprimitive functions. PhD thesis, University of Wisconsin-Madison. [640, 641]
Michael, Rothstein (1977), A new algorithm for the integration of exponential and logarithmic functions. In Proceedings of the 1977 MACSYMA Users Conference, Berkeley CA, NASA, Washington DC, 263–274. [640, 641]
John H., Rowland and John R., Cowles (1986), Small Sample Algorithms for the Identification of Polynomials. Journal of the ACM 33(4), 822–829. [199]
H., Sachse (1890), Ueber die geometrischen Isomerien der Hexamethylenderivate. Berichte der Deutschen Chemischen Gesellschaft 23, 1363–1370. [698]
H., Sachse (1892), Über die Konfigurationen der Polymethylenringe. Zeitschrift für physikalische Chemie 10, 203–241. [698]
Bruno, Salvy (1991), Asymptotique automatique et fonctions génératrices. PhD thesis, École Polytechnique, Paris. [697]
Erhard, Schmidt (1907), Zur Theorie der linearen und nichtlinearen Integralgleichungen, I. Teil: Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener. Mathematische Annalen 63, 433–476. Reprint of Erhard Schmidt's Dissertation, Göttingen, 1905. [496]
C. P., Schnorr (1982), Refined Analysis and Improvements on Some Factoring Algorithms. Journal of Algorithms 3, 101–127. [567]
C. P., Schnorr (1987), A hierarchy ofpolynomial time lattice basis reduction algorithms. Theoretical Computer Science 53, 201–224. [497]
C. P., Schnorr (1988), A More Efficient Algorithm for Lattice Basis Reduction. Journal of Algorithms 9, 47–62. [497]
C. P., Schnorr and M., Euchner (1991), Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems. In Proceedings of the 8th International Conference on Fundamentals of ComputationTheory 1991, Gosen, Germany, ed. Lothar, Budach. Lecture Notes in Computer Science 529, Springer-Verlag, 68–85. [497]
A., Schönhage (1966), Multiplikation großer Zahlen. Computing 1, 182–196. [247]
A., Schönhage (1971), Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informatica 1, 139–144. [332]
A., Schönhage (1977), Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2. Acta Informatica 7, 395–398. [245, 247, 253]
Arnold, Schönhage (1984), Factorization of univariate integer polynomials by Diophantine approximation and an improved basis reduction algorithm. In Proceedings of the 11th International Colloquium on Automata, Languages and Programming ICALP 1984, Antwerp, Belgium. Lecture Notes in Computer Science 172, Springer-Verlag, 436–447. [497]
Arnold, Schönhage (1985), Quasi-GCD Computations. Journal of Complexity 1, 118–137. [202]
A., Schönhage (1988), Probabilistic Computation of Integer Polynomial GCDs. Journal of Algorithms 9, 365–371. [202]
Arnold, Schönhage, Andreas F. W., Grotefeld, and Ekkehart, Vetter (1994), Fast Algorithms – A Multitape Turing Machine Implementation. BI Wissenschaftsverlag, Mannheim. [279, 286, 292, 727]
A., Schönhage and V., Strassen (1971), Schnelle Multiplikation großer Zahlen. Computing 7, 281–292. [221, 222, 243, 245, 247, 254, 283]
Friedrich Theodor, von Schubert (1793), De inventione divisorum. Nova Acta Academiae Scientiarum Imperalis Petropolitanae 11, 172–186. [465]
J. T., Schwartz (1980), Fast Probabilistic Algorithms for Verification of Polynomial Identities. Journal of the ACM 27(4), 701–717. [198, 332]
Štefan, Schwarz (1939), Contribution à la réductibilité des polynômes dans la théorie des congruences. Věstník Královské Ceské Společnosti Nauk, Třída Matemat.-Př Ročník Praha, 1–7. [420]
Štefan, Schwarz (1940), Sur le nombre des racines et des facteurs irréductibles d'une congruence donnée. Časopis pro pěstování matematiky a fysiky 69, 128–145. [420]
Štefan, Schwarz (1956), On the reducibility of polynomials over a finite field. Quarterly Journal of Mathematics Oxford 7(2), 110–124. [420]
ШтеΦан Шварц [Štefan, Schwarz] (1960), Об одном класе многочленов над конечным телом (On a class of polynomials over a finite field). 'Matematicko-Fyzikálny Časopis 10, 68–80. [420]
ШтеΦан Шварц [Štefan, Schwarz] (1961) О числе неприводимЫх Факторов данного многочлена над конечнЫм полем (On the number of irreducible factors of a polynomial over a finite field). Чехословачкчu мамемаuческuu мсурнал (Czechoslovak Mathematical Journal) 11(86), 213–225. [420]
Daniel, Schwenter (1636), Deliciæ Physico-Mathematicœ. Jeremias Dümler, Nürnberg. Reprint by Keip Verlag, Frankfurt am Main, 1991. [61, 131, 697]
Robert, Sedgewick and Philippe, Flajolet (1996), An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading MA. [697]
J.-A., Serret (1866), Cours d'algèbre supérieure. Gauthier-Villars, Paris, 3rd edition. [418]
Jeffrey, Shallit (1990), On the Worst Case of Three Algorithms for Computing the Jacobi Symbol. Journal of Symbolic Computation 10, 593–610. [533]
Jeffrey, Shallit (1994), Origins of the Analysis of the Euclidean Algorithm. Historia Mathematica 21, 401–419. [61]
Adi, Shamir (1979), How to Share a Secret. Communications of the ACM 22(11), 612–613. [131]
Adi, Shamir (1984), A polynomial-time algorithm for breaking the basic Merkle-Hellman cryptosystem. IEEE Transactions on Information Theory IT-30(5), 699–704. [503, 509]
A., Shamir (1993), On the Generation of Polynomials which are Hard to Factor. In Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing, San Diego CA, ACM Press, 796–804. [469]
Adi, Shamir and Richard E., Zippel (1980), On the Security of the Merkle-Hellman Cryptographic Scheme. IEEE Transactions on In formation Theory IT-26(3), 339–340. [509]
Daniel, Shanks and John W., Wrench Jr. (1962), Calculation of π to 100,000 Decimals. Mathematics of Computation 16, 76–99. [82]
William, Shanks (1853), Contributions to Mathematics Comprising Chiefly the Rectification of the Circle to 607 Places of Decimals. G. Bell, London. Excerpt reprinted in Berggren, Borwein & Borwein (1997), 147-161. [82, 90, 729]
C. E., Shannon (1948), A Mathematical Theory of Communication. Bell System Technical Journal 27, 379–423 and 623–656. Reprinted in Claude E. Shannon and Warren Weaver, The Mathematical Theory Of Communication, University of Illinois Press, Urbana IL, 1949. [209, 215, 307]
Shen, Kangsheng (1988), Historical Development of the Chinese Remainder Theorem. Archive of the History of Exact Sciences 38, 285–305. [131]
L. A., Shepp and S. P., Lloyd (1966), Ordered cycle lengths in a random permutation. Transactions of the American Mathematical Society 121, 340–357. [421]
Victor, Shoup (1990), On the deterministic complexity of factoring polynomials over finite fields. Information Processing Letters 33, 261–267. [421]
Victor, Shoup (1991), Topics in the theory of computation. Lecture Notes for CSC 2429, Spring term, Department of Computer Science, University of Toronto. [205]
Victor, Shoup (1994), Fast Construction of Irreducible Polynomials over Finite Fields. Journal of Symbolic Computation 17, 371–391. [421]
Victor, Shoup (1995), A New Polynomial Factorization Algorithm and its Implementation. Journal of Symbolic Computation 20, 363–397. [246, 279, 462]
Victor, Shoup (1999), Efficient Computation of Minimal Polynomials in Algebraic Extensions of Finite Fields. In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation ISSAC ′99, Vancouver, Canada, ed. Sam, Dooley, ACM Press, 53–58. [354]
Igor E., Shparlinski (1992), Computational and Algorithmic Problems in Finite Fields. Mathematics and Its Applications 88, Kluwer Academic Publishers. [419]
Igor E., Shparlinski (1999), Finite Fields: Theory and Computation. Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht/Boston/London. [419]
Amir, Shpilka and Amir, Yehudayoff (2010), Arithmetic Circuits: a survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science 5(3–4), 207–388. [199]
M., Sieveking (1972), An Algorithm for Division of Powerseries. Computing 10, 153–156. [286]
Joseph H., Silverman (1986), The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106, Springer-Verlag, New York. [568]
Robert D., Silverman (1987), The Multiple Polynomial Quadratic Sieve. Mathematics of Computation 48(177), 329–339. [567]
Michael F., Singer (1991), Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients. Journal of Symbolic Computation 11, 251–273. [641]
Simon, Singh (1997), Fermat's Enigma: The epic quest to solve the world's greatest mathematical problem. Anchor Books, New York. [514]
Michael, Sipser (1997), Introduction to the Theory of Computation. PWS Publishing Company, Boston MA. [89, 721]
A. O., Slisenko (1981), Complexity problems in computational theory. Ycnexu Mame mamuqecku Hayk (Uspekhi Matematicheski Nauk) 36(6), 21–103. Russian Mathematical Surveys 36 (1981), 23-125. [419]
R., Solovay and V., Strassen (1977), A fast Monte-Carlo test for primality. SIAM Journal on Computing 6(1), 84–85. Erratum in 7 (1978), p. 118. [198, 529, 530, 533]
Jonathan P., Sorenson (1998), Trading Time for Space in Prime Number Sieves. In Algorithmic Number Theory, Third International Symposium, ANTS-III, Portland, Oregon, USA, ed. J. P., Buhler. Lecture Notes in Computer Science 1423, Springer-Verlag, Berlin, Heidelberg, 179–195. [533]
В.Г. СпринѬкук (1981), ДиоФантовЫ уравенения с неизвестными простыми числами. Труәы Мамемамuцескоѕо uнсмuмума АН СССР 158, 180–196. V. G., Sprindzhuk, Diophantine equations with unknown prime numbers, Proc. Steklov Institute of Mathematics 158 (1983), 197–214. [498]
V. G., Sprindžuk (1983), Arithmetic specializations in polynomials. Journal für die reine und angewandte Mathematik 340, 26–52. [498]
J., Stein (1967), Computational Problems Associated with Racah Algebra. Journal of Computational Physics 1, 397–405. [61]
P., Stevenhagen and H. W., Lenstra Jr. (1996), Chebotarëv and his density theorem. The Mathematical Intelligencer 18(2), 26–37. [441]
Simon, Stevin (1585), De Thiende. Christoffel Plantijn, Leyden. Übersetzt und erläutert von Helmuth Gericke und Kurt Vogel, Akademische Verlagsgesellschaft, Frankfurt am Main, 1965. [41, 61]
Jacobus (James), Stirling (1730), Methodus Differential: sive Tractatus de Summatione et Interpolation Serierum Infinitarum. Gul. Bowyer, London. Translated into English with the Author's Approbation By Francis Holliday, Master of the Grammar Free-School at Haughton-Park near Retford, Nottinghamshire, London, 1749. [670]
Arne, Storjohann (1996), Faster Algorithms for Integer Lattice Basis Reduction. Technical Report 249, Eidgenössische Technische Hochschule Zürich. 24 pages, ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/2xx/249.ps.gz. [497]
Arne, Storjohann (2000), Algorithms for Matrix Canonical Forms. PhD thesis, Swiss Federal Institute of TechnologyZürich. [353]
Gilbert, Strang (1980), Linear Algebra and Its Applications. Academic Press, New York, second edition. [713]
Volker, Strassen (1969), Gaussian Elimination is not Optimal. Numerische Mathematik 13, 354–356. [335, 337, 352]
V., Strassen (1972), Berechnung und Programm. I. Acta Informatica 1, 320–335. [497]
Volker, Strassen (1973a), Vermeidung von Divisionen. Journal für die reine und angewandte Mathematik 264, 182–202. [286, 352, 497]
V., Strassen (1973b), Berechnung und Programm. II. Acta Informatica 2, 64–79. [497]
Volker, Strassen (1976), Einige Resultate über Berechnungskomplexität. Jahresberichte der DMV 78, 1–8. [541, 567]
V., Strassen (1983), The computational complexity of continued fractions. SIAM Journal on Computing 12(1), 1–27. [324, 332]
Volker, Strassen (1984), Algebraische Berechnungskomplexität. In Perspectives in Mathematics, Anniversary of Oberwolfach 1984, 509–550. Birkhäuser Verlag, Basel. [352]
Volker, Strassen (1990), Algebraic Complexity Theory. In Handbook of Theoretical Computer Science, vol. A, ed. J., van Leeuwen, 633–672. Elsevier Science Publishers B.V., Amsterdam, and The MIT Press, Cambridge MA. [352]
C., Sturm (1835), Mémoire sur la résolution des équations numériques. Mémoires présentés par divers savants à l'Acadèmie des Sciences de l'Institut de France 6, 273–318. [94]
Antonin, Svoboda (1957), Rational numerical system of residual classes. Stroje na Zpracování Informací, Sborník V, Nakl. ČSAV 5, 9–37. [132]
Antonin, Svoboda and Miroslav, Valach (1955), Operatorové obvody (Operational Circuits). With summaries in Russian and English. Stroje na Zpracování Informací 3, 247–295. [132]
Richard G., Swan (1962), Factorization of polynomials over finite fields. Pacific Journal of Mathematics 12, 1099–1106. [207, 332]
J. J., Sylvester (1840), A method of determining by mere inspection the derivatives from two equations of any degree. Philosophical Magazine 16, 132–135. Mathematical Papers 1, Chelsea Publishing Co., New York, 1973, 54–57. [197, 199]
J. J., Sylvester (1853), On the explicit values of Sturm's quotients. Philosophical Magazine VI, 293–296.
Mathematical Papers 1, Chelsea Publishing Co., New York, 1973, 637–640. [197, 727]
J. J., Sylvester (1881), On the resultant of two congruences. Johns Hopkins University Circulars 1, p. 131. Mathematical Papers 3, Chelsea Publishing Co., New York, 1973, p. 475. [197]
Nicholas S., Szabó and Richard I., Tanaka (1967), Residue arithmetic and its applications to computer technology. McGraw-Hill, New York. [132]
G., Tarry (1898), Question 1401. Le problème chinois. L'Intermédiaire des Mathématiciens 5, 266–267. Solution by Korselt. [531]
Alfred, Tarski (1948), A decision method for elementary algebra and geometry. The Rand Corporation, Santa Monica CA, 2nd edition. Project Rand, R-109. [619]
Brook, Taylor (1715), Methodus Incrementorum Directa & Inversa. Gul. Innys, London. [286]
Richard, Taylor and Andrew, Wiles (1995), Ring-theoretic properties of certain Hecke algebras. Annals of Mathematics 141, 553–572. [514]
Gérald, Tenenbaum (1995), Introduction to analytic and probabilistic number theory. Cambridge studies in advanced mathematics 46, Cambridge University Press, Cambridge, UK. [536]
A., Thue (1902), Et par andtydninger til en talteoretisk methode. Videnskabers Selskab Forhandlinger Christiana 7. [132]
А.Л. Тоом (1963), О сложности схемы из функцоинальных әлементов, реализирующей умножене целых чисел. ДоклаӘы Акаәемuu Наук СССР 150(3), 496-498. A. L., Toom, The complexity of a scheme of functional elements realizing the multiplication of integers, Soviet Mathematics Doklady 4 (1963), 714–716. [247]
Barry M., Trager (1976), Algebraic Factoring and Rational Function Integration. In Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation SYMSAC ′76, Yorktown Heights NY, ed. R. D., Jenks, ACM Press, 219–226. [466, 640]
Carlo, Traverso (1988), Gröbner trace algorithms. In Proceedings of the 1988 International Symposium on Symbolic and Algebraic Computation ISSAC ′88, Rome, Italy, ed. P., Gianni. Lecture Notes in Computer Science 358, Springer-Verlag, Berlin, 125–138. [619]
Johannes, Tropfke (1902), Geschichte der Elementar-Mathematik, vol. 1. Veit & Comp., Leipzig. [88]
Nicola, Trudi (1862), Teoria de' determinanti e loro applicazioni Libreria Scientifica e Industriale de B. Pellerano, Napoli. [199]
A. M., Turing (1937), On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Second Series, 42, 230–265, and 43, 544–546. [419]
Christopher, Umans (2008), Fast Polynomial Factorization and Modular Composition in Small Characteristic. In Proceedings of the Fourtieth Annual ACM Symposium on Theory of Computing, Victoria, BC, Canada, ACM Press, 481–490. Invited to the STOC 2008 special issue of SICOMP. [339, 751]
Alasdair, Urquhart (1995), The complexity of propositional proofs. The Bulletin of Symbolic Logic 1(4), 425–467. [697]
Giovanni, Vacca (1894), Intorno alla prima dimostrazione di un teorema di Fermat. Bibliotheca Mathematica, Serie 2, 8, 46–48. [88]
Brigitte, Vallée (2003), Dynamical Analysis of a Class of Euclidean Algorithms. Theoretical Computer Science 297, 447–486. [61]
Ch.-J. de la Vallée, Poussin (1896), Recherches analytiques sur la théorie des nombres premiers. Annales de la Société Scientifique de Bruxelles 20, 183–256 and 281–397. [533]
R. C., Vaughan (1974), Bounds for the coefficients of cyclotomic polynomials. Michigan Mathematical Journal 21, 289–295. [198]
G. S., Vernam (1926), Cipher Printing Telegraph Systems. Journal of the American Institute of Electrical Engineers 45, 109–115. [580]
G., Villard (1997), Further Analysis of Coppersmith's Block Wiedemann Algorithm for the Solution of Sparse Linear Systems. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ISSAC ′97, Maui HI, ed. Wolfgang W., Küchlin, ACM Press, 32–39. [353]
Jeffrey Scott, Vitter and Philippe, Flajolet (1990), Average-Case Analysis of Algorithms and Data Structures. In Handbook of Theoretical Computer Science, vol. A, ed. J., van Leeuwen, 431–524. Elsevier Science Publishers B.V., Amsterdam, and The MIT Press, Cambridge MA. [697]
L. G., Wade Jr. (1995), Organic Chemistry. Prentice-Hall, Inc., Englewood Cliffs NJ, 3rd edition. [698]
Bartel L., van der Waerden (1930a), Eine Bemerkung über die Unzerlegbarkeit von Polynomen. Mathematische Annalen 102, 738–739. [419]
B. L., van der Waerden (1930b), Moderne Algebra, Erster Teil. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 33, Julius Springer, Berlin. English translation: Algebra, Volume I, Springer Verlag, 1991. [586, 703]
B. L., van der Waerden (1931), Moderne Algebra, Zweiter Teil. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 34, Julius Springer, Berlin. English translation: Algebra, Volume II., Springer Verlag, 1991. [349, 586, 703]
B. L., van der Waerden (1934), Die Seltenheit der Gleichungen mit Affekt. Mathematische Annalen 109, 13–16. [465]
B. L., van der Waerden (1938), Eine Bemerkung zur numerischen Berechnung von Determinanten und Inversen von Matrizen. Jahresberichte der DMV 48, 29–30. [352]
Samuel S., Wagstaff Jr. (1983), Divisors of Mersenne numbers. Mathematics of Computation 40(161), 385–397. [534]
Gregory K., Wallace (1991), The JPEG Still Picture Compression Standard. Communications of the ACM 34(4), 30–44. [368]
D., Wan (1993), A p-adic lifting lemma and its applications to permutation polynomials. In Proceedings 1992 Conference on Finite Fields, CodingTheory, and Advancesin Communications and Computing, eds. G. L., Mullen and P. J.-S., Shiue. Lecture Notes in Pure and Applied Mathematics 141, Marcel Dekker, Inc., 209–216. [425]
Xinmao, Wang and Victor Y., Pan (2003), Acceleration of Euclidean algorithm and rational number reconstruction. SIAM Journal on Computing 32(2), 548–556. [327]
Edward, Waring (1770), Meditationes Algebraicx. J. Woodyer, Cambridge, England, second edition. English translation by Dennis Weeks, American Mathematical Society, 1991. [286]
Edward, Waring (1779), Problems concerning Interpolations. Philosophical Transactions of the Royal Society of London 69(7), 59–67. [131]
Stephen M., Watt and Hans J., Stetter, eds. (1998), Symbolic-Numeric Algebra for Polynomials. Special Issue of the Journal of Symbolic Computation 26(6). [41]
Ingo, Wegener (1987), The Complexity of Boolean Functions. Wiley-Teubner Series in Computer Science, B. G. Teubner, Stuttgart, and John Wiley & Sons. [721]
B. M. M., de Weger (1989), Algorithms for Diophantine equations. CWI Tract no. 65, Centrum voor Wiskunde en Informatica, Amsterdam. 212 pages. [497]
André, Weil (1984), Number theory: An approach through history; From Hammurapi to Legendre. Birkhäuser Verlag. xxi+375 pages. [513]
André, Weilert (2000), (1 + i)-ary GCD Computation in Z[i] as an Analogue to the Binary GCD Algorithm. Journal of Symbolic Computation 30(5), 605–617. [61]
Andreas, Werckmeister (1691), Musicalische Temperatur. Theodorus Philippus Calvisius, Franckfurt und Leipzig. First edition 1686/87. Reprint edited by Guido Bimberg and Rüdiger Pfeiffer, Denkmäler der Musik in Mitteldeutschland: Ser. 2., Documenta theoretica musicae; Bd. 1: Werckmeister-Studien. Verlag Die Blaue Eule, Essen, 1996. [86]
Douglas H., Wiedemann (1986), Solving Sparse Linear Equations Over Finite Fields. IEEE Transactions on Information Theory IT-32(1), 54–62. [340, 346, 351, 352, 355, 556]
Andrew, Wiles (1995), Modular elliptic curves and Fermat's Last Theorem. Annals of Mathematics 142, 443–551. [514]
Herbert S., Wilf (1994), generatingfunctionology. Academic Press, 2nd edition. First edition 1990. [466, 697]
Herbert S., Wilf and Doron, Zeilberger (1990), Rational functions certify combinatorial identities. Journal of the American Mathematical Society 3(1), 147–158. [697]
Herbert S., Wilf and Doron, Zeilberger (1992), An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Inventiones mathematicae 108, 575–633. [671, 697]
Michael, Willett (1978), Factoring polynomials over a finite field. SIAM Journal on Applied Mathematics 35, 333–337. [419]
H. C., Williams (1982), A p + 1 Method of Factoring. Mathematics of Computation 39(159), 225–234. [568]
H. C., Williams (1993), How was F6 factored?Mathematics of Computation 61(203), 463–474. [542]
H. C., Williams and Harvey, Dubner (1986), The primality of R1031. Mathematics of Computation 47(176), 703–711. [530]
H. C., Williams and M. C., Wunderlich (1987), On the Parallel Generation of the Residues for the Continued Fraction Factoring Algorithm. Mathematics of Computation 48(177), 405–423. [569]
Leland H., Williams (1961), Algebra of Polynomials in Several Variables for a Digital Computer. Journal of the ACM 8, 29–40. [20]
Virginia Vassilevska, Williams (2011), Breaking the Coppersmith-Winograd barrier. http://www.cs.berkeley.edu/~virgi/.Last visited 08 December 2011. 72 pp. [352]
S., Winograd (1971), On Multiplication of 2 × 2 matrices. Linear Algebra and its Applications 4, 381–388. [352]
Wen-Tsün, Wu (1994), Mechanical Theorem Proving in Geometries: Basic Principles. Texts and Monographs in Symbolic Computation, Springer-Verlag, Wien and New York. English translation by Xiaofan Jin and dongming Wang. Originally published as “Basic Principles of Mechanical Theorem Proving in Geometry” in Chinese language by Science Press, Beijing, 1984, XIV and 288 pp. [618, 619]
Chee K., Yap (1991), A New Lower Bound Construction for Commutative Thue Systems with Applications. Journal of Symbolic Computation 12, 1–27. [618]
Alexander J., Yee and Shigeru, Kondo (2011), Pi – 10 Trillion Digits. Last visited 16 October 2011. [90]
David Y. Y., Yun (1976), On Square-free Decomposition Algorithms. In Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation SYMSAC ′76, Yorktown Heights NY, ed. R. D., Jenks, ACM Press, 26–35. [419, 466]
David Y. Y., Yun (1977a), Fast algorithm for rational function integration. In Information Processing 77—Proceedings of the IFIP Congress 77, ed. B., Gilchrist, North-Holland, Amsterdam, 493–498. [640]
David Y. Y., Yun (1977b), On the equivalence of polynomial gcd and squarefree factorization problems. In Proceedings of the 1977 MACSYMA Users Conference, Berkeley CA, NASA, Washington DC, 65–70. [425]
Hans, Zassenhaus (1969), On Hensel Factorization, I. Journal of Number Theory 1, 291–311. [417, 444, 466]
Doron, Zeilberger (1990a), A holonomic systems approach to special function identities. Journal of Computational and Applied Mathematics 32, 321–368. [671, 697]
Doron, Zeilberger (1990b), A fast algorithm for proving terminating hypergeometric identities. Discrete Mathematics 80, 207–211. [671, 697]
Doron, Zeilberger (1991), The Method of Creative Telescoping. Journal of Symbolic Computation 11, 195–204. [671, 697]
Doron, Zeilberger (1993), Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture. Notices of the American Mathematical Society 40(8), 978–981. [697]
Paul, Zimmermann (1991), Séries génératrices et analyse automatique d'algorithmes. PhD thesis, École Polytechnique, Paris. [697]
Philip R., Zimmermann (1996), The Official PGP User's Guide. MIT Press. [18]
Richard, Zippel (1979), Probabilistic Algorithms for sparse Polynomials. In Proceedings of EUROSAM ′79, Marseille, France. Lecture Notes in Computer Science 72, Springer-Verlag, 216–226. [198, 498]
Richard, Zippel (1993), Effective polynomial computation. Kluwer Academic Publishers, Boston MA. [204]