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Published online by Cambridge University Press:  05 July 2016

David Masser
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Universität Basel, Switzerland
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References

Adolphson, A.C., Conrey, J.B., Ghosh, A., Yager, R.I. (eds.) (1987). Analytic Number Theory and Diophantine Problems (Stillwater 1984), Progress in Math. 70, Birkhäuser.
Amoroso, F., David, S. (1999). Le probléme de Lehmer en dimension supérieure, J. reine angew. Math. 513, 145–179.Google Scholar
Amoroso, F., Dvornicich, R. (2000). A lower bound for the height in abelian extensions, J. Number Theory 80, 260–272.Google Scholar
Amoroso, F., Mignotte, M. (1996). On the distribution of the roots of polynomials, Ann. Inst. Fourier, Grenoble 46, 1275–1291.Google Scholar
Amoroso, F., Zannier, U. (2000). A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29, 711–727.Google Scholar
Amoroso, F., Zannier, U. (eds.) (2003). Diophantine Approximation (Cetraro Italy 2000), Lecture Notes in Math. 1819, Springer.
André, Y. (1989). G-Functions and Geometry, Aspects of Math. E13, Vieweg.
Baker, H.F. (1907). An Introduction to the Theory of Multiply Periodic Functions, Cambridge University Press.
Baker, A. (1964). Rational approximations to and other algebraic numbers, Quarterly J. Math. 15, 375–383.Google Scholar
Baker, A. (1966). Linear forms in the logarithms of algebraic numbers I, Mathematika 13, 204–216.Google Scholar
Baker, A. (1967a). Linear forms in the logarithms of algebraic numbers II, Mathematika 14, 102–107.Google Scholar
Baker, A. (1967b). Linear forms in the logarithms of algebraic numbers III, Mathematika 14, 220–228.Google Scholar
Baker, A. (1968a). Linear forms in the logarithms of algebraic numbers IV, Mathematika 15, 204–216.Google Scholar
Baker, A. (1968b). Contributions to the theory of diophantine equations. II. The diophantine equation, Phil. Trans. Roy. Soc. London 263A, 193–208.Google Scholar
Baker, A. (ed.) (1988). New Advances in Transcendence Theory, Cambridge University Press.
Baker, A. (1990). Transcendental Number Theory, Cambridge University Press.
Baker, A. (2012). A Comprehensive Course in Number Theory, Cambridge University Press.
Baker, A., Masser, D. (eds.) (1977). Transcendence Theory: Advances and Applications, Academic Press.
Baker, A., Wüstholz, G. (1993). Logarithmic forms and group varieties, J. reine angew. Math. 442, 19–62.Google Scholar
Baker, A., Wüstholz, G. (2007). Logarithmic Forms and Diophantine Geometry, Cambridge University Press.
Barré, K., Diaz, G., Gramain, F., Philibert, G. (1996). Une preuve de la conjecture de Mahler– Manin, Inventiones Math. 124, 1–9.Google Scholar
Bays, M., Habegger, P. (2015). A note on divisible points on curves, Trans. Amer. Math. Soc. 367, 1313–1328.Google Scholar
Bertrand, D., Masser, D. (1980). Linear forms in elliptic logarithms, Inventiones Math. 58, 283–288.Google Scholar
Bertrand, D., Waldschmidt, M. (eds.) (1983). Approximations Diophantiennes et Nombres Transcendants (Luminy 1982), Progress in Math. 31, Birkhäuser.
Beukers, F. (1979). A note on the irrationality of ξ(2) and ξ(3), Bull. London Math. Soc. 11, 268–272.Google Scholar
Beukers, F. (1997). On a sequence of polynomials, J. Pure Appl. Algebra 117/118, 97–103.Google Scholar
Bilu, Yu.F. (1997) Limit distribution of small points on algebraic tori, Duke Math. J. 89, 465–476.Google Scholar
Bilu, Yu.F., Masser, D. (2006). A quick proof of Sprindzhuk's decomposition theorem. In More Sets, Graphs and Numbers – A Salute to Vera Sòs and András Hajnal (eds. E., Györi, G., Katona, L., Lovász), Bolyai Society Mathematical Studies 15, Springer (pp. 25–32).
Bilu, Yu.F., Bugeaud, Y., Mignotte, M. (2014). The Problem of Catalan, Springer.
Bilu, Yu.F., Masser, D., Zannier, U. (2013). An effective “Theorem of André” for CM-points on a plane curve, Math. Proc. Cambridge Phil. Soc. 154, 145–152.Google Scholar
Boas, R.P. (1954). Entire Functions, Academic Press.
Bombieri, E. (1970). Algebraic values of meromorphic maps, Inventiones Math. 10, 267–287; 11, 163–166.Google Scholar
Bombieri, E. (1982). On the Thue–Siegel–Dyson theorem, Acta Math. 148, 255–296.Google Scholar
Bombieri, E. (1983). On Weil's “Théorème de décomposition”, Amer. J. Math. 105, 295–308.Google Scholar
Bombieri, E., Gubler, W. (2006). Heights in Diophantine Geometry, Cambridge University Press.
Bombieri, E., Pila, J. (1989). The number of integral points on arcs and ovals, Duke Math. J. 59, 337–357.Google Scholar
Bombieri, E., Vaaler, J.D. (1983). On Siegel's Lemma, Inventiones Math. 73, 11–32.Google Scholar
Bombieri, E., Zannier, U. (1995). Algebraic points on subvarieties of Gn m, Int. Math. Res. Notices 7, 333–347.Google Scholar
Bombieri, E., Masser, D., Zannier, U. (1999). Intersecting a curve with algebraic subgroups of multiplicative groups, Int. Math. Research Notices 20, 1119–1140.Google Scholar
Boyd, D.W. (1992). Two sharp inequalities for the norm of a factor of a polynomial, Mathematika 39, 341–349.Google Scholar
Brenner, J.L., Foster, L.L. (1982). Exponential diophantine equations, Pacific. J. Math. 101, 263–301.Google Scholar
Brownawell, W.D. (1974). The algebraic independence of certain numbers related by the exponential function, J. Number Theory 6, 22–31.Google Scholar
Brownawell, W.D. (1983). Zero estimates for solutions of differential equations. In Bertrand and Waldschmidt (1983) (pp. 67–94).Google Scholar
Brownawell, W.D., Masser, D. (1980) Multiplicity estimates for analytic functions I, J. reine angew. Math. 314, 200–216.Google Scholar
Buck, R.C. (1948). Integral valued entire functions, Duke Math. J. 15, 879–891.Google Scholar
Burger, E.B., Tubbs, R. (2004). Making Transcendence Transparent, Springer.
Carlitz, L. (1935). On certain functions connected with polynomials in a Galois field, Duke Math. J. 1, 137–168.Google Scholar
Cassels, J.W.S. (1965). An Introduction to Diophantine Approximation, Tracts in Mathematics and Mathematical Physics 45, Cambridge University Press.
Cassels, J.W.S. (1991). Lectures on Elliptic Curves, LMS Student Texts 24, Cambridge University Press.
Chowla, S., Selberg, A. (1967). On Epstein's zeta function, J. reine angew. Math. 227, 86–110.Google Scholar
Chudnovsky, G.V. (1984). Contributions to the Theory of Transcendental Numbers, Math. Surveys and Monographs 19, Amer. Math. Soc.
Cohen, P.B. (1984). On the coefficients of the transformation polynomials for the elliptic modular function, Math. Proc. Camb. Phil. Soc. 95, 389–402.Google Scholar
Cohen, P.B., Zannier, U. (2000) Multiplicative dependence and bounded height, an example. In Proc. Algebraic Number Theory and Diophantine Analysis Conference (Graz 1998), Walter de Gruyter (pp. 93–101).
Conway, J.H., Guy, R.K. (1996). The Book of Numbers, Springer.
Corvaja, P., Zannier, U. (2002). Some new applications of the subspace theorem, Compositio Math. 131, 319–340.Google Scholar
Corvaja, P., Zannier, U. (2013). Greatest common divisors of u − 1, v − 1 in positive characteristic and rational points on curves over finite fields, J. Eur. Math. Soc. 15, 1927–1942 Google Scholar
David, S., Philippon, P. (1999). Minorations des hauteurs normalisées des sous-variétés des tores, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28, 489–543.Google Scholar
Dixon, A.L. (1908). The eliminant of three quantics in two independent variables, Proc. London Math. Soc. 6, 468–478.Google Scholar
Dobrolowski, E. (1979). On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34, 391–401.Google Scholar
Duverney, D. (1997). Sur les propriétés arithmétiques de la fonction de Tschakaloff, Periodica Math. Hungarica 35, 149–157.Google Scholar
Dyson, F.J. (1947). The approximation to algebraic numbers by rationals, Acta Math. 79, 225–240.Google Scholar
Eagle, A. (1958). The Elliptic Functions as They Should Be, Galloway and Porter.
Eichler, M. (1966). Introduction to the Theory of Algebraic Numbers and Functions, Academic Press.
Erdʺos, P., Turán, P. (1950). On the distribution of roots of polynomials, Annals of Math. 51, 105–119.Google Scholar
Esnault, H., Viehweg, E. (1984). Dyson's Lemma for polynomials in several variables (and the theorem of Roth), Inventiones Math. 78, 445–490.Google Scholar
Evertse, J.H., Silverman, J.H. (1986). Uniform bounds for the number of solutions to, Math. Proc. Cambridge Phil. Soc. 100, 237–248.Google Scholar
Favre, C., Rivera-Letelier, J. (2006). Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Annalen 335, 311–361.Google Scholar
Feldman, N.I. (1951). Approximation of certain transcendental numbers. II: The approximation of certain numbers connected with the Weierstrass -function, Izv. Akad. Nauk. SSSR 15, 153–176 (Russian); English transl. Amer. Math. Soc. Trans. 59 (1966), 246–270.Google Scholar
Feldman, N.I. (1971). An effective refinement of the exponent in Liouville's theorem, Izv. Akad. Nauk. SSSR 35, 973–990 (Russian); English transl. Math. USSR Izv. 5, 985–1002.Google Scholar
Feldman, N.I., Nesterenko, Yu.V. (1997). Number Theory IV, Transcendental Numbers, Encyclopaedia of Math. Sci. 44 (eds. A.N., Parshin, I.R., Shafarevich), Springer.
Galochkin, A.I. (1974). Lower bounds of polynomials in the values of a certain class of analytic functions, Mat. Sbornik 95, 396–417 (Russian); English transl. Math. USSR Sb. 24, 385–407.Google Scholar
Galochkin, A.I. (1980). Transcendence measure of values of functions satisfying certain functional equations, Mat. Zametki 27, 175–183 (Russian); English transl. Math. notes Acad. Sci. USSR 27, 83–88.Google Scholar
Garcia, A., Voloch, J.F. (1988). Fermat curves over finite fields, J. Number Theory 30, 345–356.Google Scholar
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V. (1994). Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser.
Gelfond, A.O. (1934). Sur quelques résultats nouveaux dans la théorie des nombres transcendants, Comptes Rendus Acad. Sci. Paris, 199, 259.Google Scholar
Gelfond, A.O. (1960). Transcendental and Algebraic Numbers, Dover.
Ghosh, A., Ward, K. (2015). The number of roots of polynomials in a prime field, Int. Math. Res. Notices 4, 898–926.Google Scholar
Goss, D. (1996). Basic Structures of Function Field Arithmetic, Ergebnisse Math. 35, Springer.
Gramain, F. (1978). Fonctions entiéres arithmétiques. In Séminaire P. Lelong–H. Skoda (Analyse) 1976/77, Lecture Notes in Math. 694, Springer (pp. 96–125).
Gramain, F. (1981). Sur le théorème de Fukasawa–Gelfond, Inventiones Math. 63, 495–506.Google Scholar
Gramain, F., Mignotte, M., Waldschmidt, M. (1986). Valeurs algébriques de fonctions analytiques, Acta Arith. 47, 97–121.Google Scholar
Grant, D. (1990). Formal groups in genus two, J. reine angew. Math. 411, 96–121.Google Scholar
Greenhill, A.G. (1892). The Applications of Elliptic Functions, Macmillan.
Grytczuk, A., Schinzel, A. (1991). On Runge's theorem about diophantine equations. In Coll. Math. Soc. János Bolyai 60 (Sets, Graphs and Numbers), North-Holland (pp. 329–356).
Habegger, P. (2005). The equation in multiplicatively independent unknowns, Acta Arith. 119, 349–372.Google Scholar
Habegger, P. (2007). Heights and Multiplicative Relations on Algebraic Varieties, Doctoral Thesis, Basle.
Hall, R.R. (1971). On pseudo-polynomials, Mathematika 18, 71–77.Google Scholar
Halphen, G.H. (1886). Traité des Fonctions Elliptiques et de leurs Applications, I. Gauthier-Villars.
Hančl, J., Tijdeman, R. (2004). On the irrationality of Cantor series, J. reine angew. Math. 571, 145–158.Google Scholar
Heath-Brown, D.R. (1996). An estimate for Heilbronn's exponential sum. In Analytic Number Theory (Proceedings of a Conference in Honor of Heini Halberstam Volume 2), Progr. Math. 139, Birkhäuser (eds. B.C., Berndt, H.G., Diamond, A.J., Hildebrand) (pp. 451–463).
Heath-Brown, D.R., Konyagin, S. (2000). New bounds for Gauss sums derived from kth powers, and for Heilbronn's exponential sum, Quart. J. Math. 51, 221–235.Google Scholar
Hensel, K. (1905). Über die arithmetischen Eigenschaften der algebraischen und transzendenten Zahlen, Jahresbericht der Deutschen Mathematiker-Vereinigung 14, 545–558.Google Scholar
Hilliker, D.L., Straus, E.G. (1983). Determination of bounds for the solutions to those binary diophantine equations that satisfy the hypotheses of Runge's theorem, Trans. Amer. Math. Soc. 280, 637–657.Google Scholar
Hindry, M., Silverman, J.H. (2000). Diophantine Geometry, Graduate Texts in Math. 201, Springer.
Jones, G.O., Wilkie, A.J. (eds.) (2015). O-minimality and Diophantine Geometry, LMS Lecture Note Series 421, Cambridge University Press.
Klein, F. (1956). Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Dover.
Kmošek, M. (1979). Continued Fraction Expansion of some Irrational Numbers, Master's thesis, Uniwersytet Warszawski (in Polish).
Kowalski, E. (2010). Lecture notes and related documents, available online at www.math.ethz.ch/kowalski/exp-sums.html
Kronecker, L. (1857). Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. reine angew. Math. 53, 173–175.Google Scholar
Kühne, L. (2013). An effective result of André–Oort type II, Acta Arith. 161, 1–19.Google Scholar
Lang, S. (1996a). Introduction to Transcendental Numbers, Addison-Wesley.
Lang, S. (1996b). Introduction to Diophantine Approximations, Addison-Wesley.
Lang, S. (1973). Elliptic Functions, Addison-Wesley.
Lang, S. (1978). Elliptic Curves, Diophantine Analysis, Grundlehren der Math. Wiss. 231, Springer.
Lang, S. (1983). Fundamentals of Diophantine Geometry, Springer.
Lang, S. (1993). Algebra, Addison-Wesley.
Laurent, M. (1983).Minoration de la hauteur de Néron–Tate. In Séminaire de théorie des nombres, Paris 1981/1982, Progr. Math 38, Birkhäuser (ed. M.-J., Bertin) (pp. 137–151).
Lehmer, D.H. (1933). Factorization of certain cyclotomic functions, Annals of Math. 34, 461–479.Google Scholar
Leitner, D.J. (2011). Two exponential diophantine equations, J. Théorie des Nombres Bordeaux 23, 479–487.Google Scholar
Loher, T., Masser, D. (2004). Uniformly counting points of bounded height, Acta Arithmetica 111, 277–297.Google Scholar
Loxton, J.H., van der Poorten, A.J. (1978). Algebraic independence properties of the Fredholm series, J. Australian Math. Soc. A26, 31–45.Google Scholar
Macaulay, F.S. (1916). The Algebraic Theory of Modular Systems, Tracts in Mathematics and Mathematical Physics 19, Cambridge University Press.
Mahler, K. (1953). On the approximation of π , Koninkl. Nederl. Akad. Wet. A56, 30–42 or Indagationes Math. 15, 30–42.Google Scholar
Mahler, K. (1964). Inequalities for ideal bases in algebraic number fields, J. Australian Math. Soc. 4, 425–448.Google Scholar
Mahler, K. (1968). Applications of a theorem by A.B., Shidlovsky, Phil. Trans. Roy. Soc. London A305, 149–173.
Mahler, K. (1975). On the transcendency of the solutions of a special class of functional equations, Bull. Australian Math. Soc. 13, 389–410.Google Scholar
Mahler, K. (1976). Lectures on Transcendental Numbers, Lecture Notes in Math. 546, Springer.
Masser, D. (1975). Elliptic Functions and Transcendence, Lecture Notes in Math. 437, Springer.
Mahler, K. (1977). Division fields of elliptic functions, Bull. London Math. Soc. 9, 49–53.Google Scholar
Mahler, K. (1980a). Sur les fonctions entières à valeurs entières, Comptes Rendus Acad. Sci. Paris 291, 1–4.Google Scholar
Mahler, K. (1980b). Problem 6479, Amer. Math. Monthly 93, 486–488.
Mahler, K. (1981). A note on Baker's Theorem. In Recent Progress in Analytic Number Theory (eds. H., Halberstam and C., Hooley), Academic Press (pp. 153–158).
Mahler, K. (1983). On certain functional equations in several variables. In Bertrand and Waldschmidt (1983) (pp. 173–190).Google Scholar
Mahler, K. (2003). Sharp estimates for Weierstrass elliptic functions, J. d'Analyse Math. 90, 257–302.Google Scholar
Mahler, K. (2011). Rational values of the Riemann zeta function, J. Number Th. 131, 2037–2046.Google Scholar
Masser, D., Vaaler, J.D. (2008). Counting algebraic numbers with large height I. In Diophantine Approximation – Festschrift for Wolfgang Schmidt (eds. H.-P., Schlickewei, K., Schmidt, R.F., Tichy), Developments in Mathematics 16, Springer (pp. 237–243).
Masser, D., Wolbert, J. (1993). On products of polynomials, Proc. Amer. Math. Soc. 117, 593–599.Google Scholar
Masser, D., Wüstholz, G. (1986). Algebraic independence of values of elliptic functions, Math. Annalen 276, 1–17.Google Scholar
Masser, D., Wüstholz, G. (1990). Estimating isogenies on elliptic curves, Inventiones Math. 100, 1–24.Google Scholar
Masser, D., Zannier, U. (2010). Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132, 1677–1691.Google Scholar
Masser, D., Zannier, U. (2015). Torsion points on families of abelian surfaces and Pell's equation over polynomial rings (with Appendix by V. Flynn), J. European Math. Soc. 17, 2379–2416.Google Scholar
Mignotte, M. (1979). Approximations des nombres algébriques par des nombres algébriques de grand degré, Ann. Fac. Sci Toulouse 1, 165–170.Google Scholar
Mignotte, M. (1989). Sur un théorème de M. Langevin, Acta Arith. 54, 81–86.Google Scholar
Mihăilescu, P. (2004). Primary cyclotomic units and a proof of Catalan's conjecture, J. reine angew Math. 572, 167–195.Google Scholar
Miller, W. (1982). Transcendence measures by a method of Mahler, J. Australian Math. Soc. 32, 68–78.Google Scholar
Mitkin, D.A. (1992). An estimate for the number of roots of some comparisons by the Stepanov method, Mat. Zametki 51, 52–58, 157 (Russian); English transl. Math. Notes 51, 565–570.Google Scholar
Mordell, L.J. (1923). On the integer solutions of the equation, Proc. London Math. Soc. 21, 415–419.Google Scholar
Myerson, G. (1986). How small can a sum of roots of unity be?, Amer.Math. Monthly 93, 457–459.Google Scholar
Néron, A. (1965). Quasi-fonctions et hauteurs sur les variétés abéliennes, Annals of Math. 82, 249–331.Google Scholar
Nesterenko, Yu.V. (1983). Estimates for the number of zeros of certain functions. In Bertrand and Waldschmidt (1983) (pp. 263–269).Google Scholar
Nesterenko, Yu.V. (1996). Modular functions and transcendence questions, Mat. Sb. 187/9, 65–96 (Russian); English transl. Sb. Math. 187/9, 1319–1348.Google Scholar
Nesterenko, Yu.V., Philippon, P. (eds.) (2001). Introduction to Algebraic Independence Theory, Lecture Notes in Math. 1752, Springer.
Nishioka, K. (1986). Proof of Masser's conjecture on the algebraic independence of values of Liouville series, Proc. Japan Acad. A62, 219–222.Google Scholar
Nishioka, K. (1996). Mahler Functions and Transcendence, Lecture Notes in Math. 1631, Springer.
Niven, I. (1956). Irrational Numbers, The Carus Math. Monog. 11, Math. Assoc. Amer.
Osgood, C.F. (ed.) (1973). Diophantine Approximation and its Applications, Academic Press.
Perelli, A., Zannier, U. (1981). Su un teorema di Pólya, Boll. U. Mat. Italiana 18A, 305–307.Google Scholar
Perelli, A., Zannier, U. (1984). On recurrent mod p sequences, J. reine angew. Math. 348, 135–146.Google Scholar
Petsche, C. (2005). A quantitative version of Bilu's equidistribution theorem, Int. J. Number Theory 1, 281–291.Google Scholar
Philibert, G. (1997). Un lemme de zéros modulaire, J. Number Theory 66, 306–313.Google Scholar
Philippon, P. (1983). Variétés abéliennes et indépendance algébrique II: Un analogue abélien du théorème de Lindemann–Weierstrass, Inventiones Math. 72, 389–405.Google Scholar
Philippon, P. (ed.) (1992). Approximations Diophantiennes et Nombres Transcendants (Luminy 1990), Walter de Gruyter.
Pila, J. (2004). Integer points on the dilation of a subanalytic surface, Quarterly J. Math. 55, 207–223.Google Scholar
Pila, J. (2011). O-minimality and the André–Oort conjecture for Cn , Annals of Math. 173, 1779–1840.Google Scholar
Pila, J., Wilkie, A. (2006). The rational points of a definable set, Duke Math. J. 133, 591–616.Google Scholar
Pólya, G. (1915). Über ganzwertige ganze Funktionen, Rend. Cont. Math. Palermo 40, 1–16.Google Scholar
Pontreau, C. (2005). Minoration effective de la hauteur des points d'une courbe de G2 m définie sur Q, Acta Arith. 120, 1–26.Google Scholar
van der Poorten, A.J. (1979). A proof that Euler missed… Apéry's proof of the irrationality of ξ(3), Math. Intelligencer 1, 195–203.Google Scholar
Rankin, R. (1977). Modular Forms and Functions, Cambridge University Press.
Rausch, U. (1985). On a theorem of Dobrowolski about the product of conjugate numbers, Colloq. Math. 50, 137–142.Google Scholar
Rémond, G. (2002). Sur les sous-variétés des tores, Compositio Math. 134, 337–366.Google Scholar
Roth, K.F. (1955). Rational approximations to algebraic numbers, Mathematika, 2, 1–20.Google Scholar
Runge, C. (1887). Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen, J. reine angew. Math. 100, 425–435.Google Scholar
Ruppert, W. (1986). Reduzibilität ebener Kurven, J. reine angew. Math. 369, 167–191.Google Scholar
Rusza, I.Z. (1971). On congruence preserving functions (Hungarian), Mat. Lapok, 22, 125–134.Google Scholar
Schanuel, S.H. (1979). Heights in number fields, Bull. Soc. Math. France 107, 433–449.Google Scholar
Schinzel, A. (1969). An improvement of Runge's theorem on diophantine equations, Commentarii Pontif. Acad. Sci. 2, 1–9.Google Scholar
Schinzel, A. (1982). Selected Topics on Polynomials, University of Michigan Press.
Schinzel, A. (2000). Polynomials with Special Regard to Reducibility, Encyclopaedia of Math. and its Applications 77, Cambridge University Press.
Schmidt, W.M. (1976). Equations over Finite Fields, an Elementary Approach, Lecture Notes in Math. 536, Springer.
Schmidt, W.M. (1980). Diophantine Approximation, Lecture Notes in Math. 785, Springer.
Schmidt, W.M. (1990). Eisenstein's theorem on power series expansions of algebraic functions, Acta Arith. 56, 161–179.Google Scholar
Schmidt, W.M. (1991). Diophantine Approximations and Diophantine Equations, Lecture Notes in Math. 1467, Springer.
Schneider, T. (1957). Einführung in die Transzendenten Zahlen, Grundlehren der Math. Wiss. 81, Springer.
Serre, J-P. (1988). Algebraic Groups and Class Fields, Graduate Texts in Math. 117, Springer.
Shallit, J. (1979). Simple continued fractions for some irrational numbers, J. Number Theory 11, 209–217.Google Scholar
Shanks, D. (1974). Incredible identities, Fibonacci Quart. 12, 271–280.Google Scholar
Shidlovsky, A.B. (1989). Transcendental Numbers, Walter de Gruyter.
Shkredov, I.D. (2014). On exponential sums over multiplicative subgroups of medium size, Finite Fields and Their Applications 30, 72–87.Google Scholar
Shparlinsky, I. (1991). Estimates for Gauss sums, Mat. Zametki 50, 122–130 (Russian); English transl. Math. Notes 50, 740–746.Google Scholar
Siegel, C.L. (1921). Approximation algebraischer Zahlen, Math. Zeit., 10, 173–213; Ges. Abh. I, 6–46.Google Scholar
Siegel, C.L. (1929). Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys.-math. Kl., No. 1; Ges. Abh. I, 209–266.Google Scholar
Siegel, C.L. (1935). Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1, 83–86; Ges. Abh. I, 406–409.Google Scholar
Siegel, C.L. (1949). Transcendental Numbers, Princeton.
Siegel, C.L. (1955). Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen, Math.-phys. Kl. 4, 71–77; Ges. Abh. III, 216–222.Google Scholar
Silverman, J.H. (1992). The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106, Springer.
Sprindzuk, V.G. (1993). Classical Diophantine Equations in Two Unknowns, Lecture Notes in Math. 1559, Springer.
Stark, H. (1973). Effective estimates of solutions of some diophantine equations, Acta Arith. 24, 251–259.Google Scholar
Stewart, C.L. (1978). Algebraic integers whose conjugates lie near the unit circle, Bull. Soc. Math. France 106, 169–176.Google Scholar
Stöhr, K.-O., Voloch, J.F. (1986). Weierstrass points and curves over finite fields, Proc. London Math. Soc. 52, 1–19.Google Scholar
Stolarsky, K.B. (1974). Algebraic Numbers and Diophantine Approximation, Pure and Applied Math. 26, Marcel Dekker.
Strassen, V. (1974). Polynomials with rational coefficients which are hard to compute, SIAM J. Comp. 3, 128–149.
Surroca, A. (2006). Valeurs algébriques de fonctions transcendantes, Int. Math. Research Notices ID 16834, 1–31.Google Scholar
Szpiro, L., Ullmo, E., Zhang, S. (1997). Equirépartition des petits points, Inventiones Math. 127, 337–347.Google Scholar
Thue, A. (1909). Über Annäherungswerte algebraischer Zahlen, J. reine angew. Math. 135, 284–305.Google Scholar
Thunder, J.L. (1995). Siegel's lemma for function fields, Michigan Math. J. 42, 147–162.Google Scholar
Tijdeman, R. (1971a). On the number of zeros of general exponential polynomials, Indagationes Math. 33, 1–7.Google Scholar
Tijdeman, R. (1971b). On the algebraic independence of certain numbers, Indagationes Math. 33, 146–162.Google Scholar
Viola, C. (1985). On Dyson's lemma, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12, 105–135.Google Scholar
Waldschmidt, M. (1973). Solution du huitième problème de Schneider, J. Number Theory 5, 191–202.Google Scholar
Waldschmidt, M. (1974). Nombres Transcendants, Lecture Notes in Math. 402, Springer.
Waldschmidt, M. (1978). Pólya's theorem by Schneider's method, Acta Math. Acad. Sci. Hungaricae 31, 21–25.Google Scholar
Waldschmidt, M. (1987). Nombres Transcendants et Groupes Algébriques, Astérisque 69–70.
Waldschmidt, M. (2000). Diophantine Approximation on Linear Algebraic Groups, Grundlehren der Math. Wiss. 326, Springer.
Whittaker, E.T., Watson, G.N. (1965). A Course of Modern Analysis, Cambridge University Press.
Wilkie, A.J. (2004). Diophantine properties of sets definable in o-minimal structures, J. Symbolic Logic 69, 851–861.Google Scholar
Wüstholz, G. (1983). Über das Abelsche Analogon des Lindemannschen Satzes I, Inventiones Math. 72, 363–388.Google Scholar
Wüstholz, G. (ed.) (1987). Diophantine Approximation and Transcendence Theory, Lecture Notes in Math. 1290, Springer.
Wüstholz, G. (1989a). Multiplicity estimates on group varieties, Annals of Math. 129, 471–500.Google Scholar
Wüstholz, G. (1989b). Algebraische Punkte auf analytischen untergruppen algebraischer Gruppen, Annals of Math. 129, 501–517.Google Scholar
Wüstholz, G. (ed.) (2002). A Panorama of Number Theory or the View from Baker's Garden, Cambridge University Press.
Yu, H.B. (1999). Note on Heath-Brown's estimate for Heilbronn's exponential sum, Proc. Amer. Math. Soc. 127, 1995–1998.Google Scholar
Zagier, D. (1993). Algebraic numbers close to both 0 and 1, Math. Comp. 61, 485–491.Google Scholar
Zannier, U. (1996). On periodic mod p sequences and G-functions, Manuscripta Math. 90, 391–402.Google Scholar
Zannier, U. (1998). Polynomials modulo p whose values are squares (elementary improvements on some consequences of Weil's bounds), L'Enseignement Math. 44, 95–102.Google Scholar
Zannier, U. (2003). Some Applications of Diophantine Approximation to Diophantine Equations, Forum, Udine.
Zannier, U. (ed.) (2007). Diophantine Geometry, Edizione della Normale 4 Scuola Normale Pisa.
Zannier, U. (2012). Some Problems of Unlikely Intersections in Arithmetic and Geometry (with Appendices by D. Masser), Annals of Math. Studies 181, Princeton University Press.
Zannier, U. (2014a). Lecture Notes on Diophantine Analysis, Edizione della Normale 8 Scuola Normale Pisa.
Zannier, U. (ed.) (2014b). On Some Applications of Diophantine Approximations (with Commentary by C. Fuchs and U. Zannier), Edizione della Normale 2 Scuola Normale Pisa.
Zhang, S. (1992). Positive line bundles on arithmetic surfaces, Annals of Math. 136, 569–587.Google Scholar

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  • References
  • David Masser, Universität Basel, Switzerland
  • Book: Auxiliary Polynomials in Number Theory
  • Online publication: 05 July 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781107448018.025
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  • References
  • David Masser, Universität Basel, Switzerland
  • Book: Auxiliary Polynomials in Number Theory
  • Online publication: 05 July 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781107448018.025
Available formats
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  • References
  • David Masser, Universität Basel, Switzerland
  • Book: Auxiliary Polynomials in Number Theory
  • Online publication: 05 July 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781107448018.025
Available formats
×