Akhtari, S. (2012), Representation of unity by binary forms, Trans. Amer. Math. Soc. 364, 2129–2155.
Akhtari, S. and Vaaler, J. (2015), Heights, regulators and Schinzel's determinantinequality, arXiv:1508.01969v2.
Akiyama, S., Brunotte, H. and Pethő, A. (2003), Cubic CNS polynomials, notes on aconjecture of W.J. Gilbert, J. Math. Anal. Appl. 281, 402–415.
Akiyama, S., Borbely, T., Brunotte, H., Pethő, A. and Thuswaldner, J.M. (2005), Generalized radix representations and dynamical systems I, Acta Math. Hungar. 108, 207–238.
Akizuki, S. and Ota, K. (2013), On power bases for rings of integers of relative Galoisextensions, Bull. London Math. Soc. 45, 447–452.
Amoroso, F. and Viada, E. (2009), Small points on subvarieties of a torus, Duke Math. J. 150, 407–442.
Archinard, G. (1974), Extensions cubiques cycliques de Q dont l'anneau des entiers estmonogène, Enseign. Math. 20, 179–203.
Artin, E. (1950), Questions de la base minimal dans la théorie des nombresalgébriques, Colloques Internat du Centre National Recherche Scientifique, No. 24, CNRS, Paris, pp. 19–20. Collected papers of Emil Artin, Reading (Mass.), 1965, 229–231.
Aschenbrenner, M. (2004), Ideal membership in polynomial rings over the integers, J. Amer. Math. Soc. 17, 407–442.
Baker, A. (1966), Linear forms in the logarithms of algebraic numbers, I, Mathematik. 13, 204–216.
Baker, A. (1967a), Linear forms in the logarithms of algebraic numbers, II, Mathematika
14, 102–107.
Baker, A. (1967b), Linear forms in the logarithms of algebraic numbers, III, Mathematik. 14, 220–228.
Baker, A. and Wustholz, G. (1993), Logarithmic forms and group varieties, J. Reine Angew. Math. 442, 19–62.
Barat, G., Berthe, V., Liardet, P. and Thuswaldner, J.M. (2006), Dynamical directionsin numeration, Numeration, pavages, substitutions, Ann. Inst. Fourier (Grenoble). 56, 1987–2092.
Bardestani, M. (2012), The density of a family of monogenic number fields, arXiv:1202.2047v1.
Bell, J.P. and Hare, K.G. (2009), On Z-modules of algebraic integers, Canad. J. Math. 61, 264–281.
Bell, J.P. and Hare, K.G. (2012), Corrigendum to “On Z-modules of algebraic integers,”
Canad. J. Math.
64, 254–256.
Bell, J.P. and Nguyen, K.D. (2015), Some finiteness results on monogenic orders inpositive characteristic, arXiv:1508.07624v1.
Berczes, A. (2000), On the number of solutions of index form equations, Publ. Math. Debrece. 56, 251–262.
Berczes, A., Evertse, J.-H. and Győry, K. (2004), On the number of equivalence classesof binary forms of given degree and given discriminant, Acta Arith. 113, 363–399.
Berczes, A., Evertse, J.-H. and Győry, K. (2009), Effective results for linear equationsin two unknowns from a multiplicative division group, Acta Arith. 136, 331–349.
Berczes, A., Evertse, J.-H. and Győry, K. (2013), Multiply monogenic orders, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). 12, 467–497.
Berczes, A., Evertse, J.-H. and Győry, K. (2014), Effective results for Diophantine equationsover finitely generated domains, Acta Arith. 163, 71–100.
Beresnevich, V., Bernik, V. and Gotze, F. (2010), The distribution of close conjugatealgebraic numbers, Compos. Math. 146, 1165–1179.
Beresnevich, V., Bernik, V. and Gotze, F. (2015), Integral polynomials with smalldiscriminants and resultants, arXiv:1501.05767v1.
Bernik, V., Gotze, F. and Kukso, O. (2008), Lower bounds for the number of integralpolynomials with given order of discriminant, Acta Arit. 133, 375–390.
Beukers, F. and Schlickewei, H.P. (1996), The equation x + y = 1 in finitely generatedgroups, Acta. Arith. 78, 189–199.
Bilu, Yu.F., Gaal, I. and Győry, K. (2004), Index form equations in sextic fields: a hardcomputation, Acta Arith. 115, 85–96.
Bilu, Yu.F. and Hanrot, G. (1996), Solving Thue equations of high degree, J. Number Theory. 60, 373–392.
Bilu, Yu.F. and Hanrot, G. (1999), Thue equations with composite fields, Acta Arith.. 88, 311–326.
Birch, B.J. and Merriman, J.R. (1972), Finiteness theorems for binary forms with givendiscriminant, Proc. London Math. Soc. 24, 385–394.
Bombieri, E. and Gubler, W. (2006), Heights in Diophantine Geometry, Cambridge University Press.
Bombieri, E. and Vaaler, J. (1983), On Siegel's/lemma, Invent. Math. 73, 11–32.
Borevich, Z.I. and Shafarevich, I.R. (1967), Number Theory, 2nd ed., Academic Press.
Borosh, I., Flahive, M., Rubin, D. and Treybig, B. (1989), A sharp bound for solutionsof linear Diophantine equations, Proc. Amer. Math. Soc. 105, 844–846.
Bosma, W., Cannon, J. and Playoust, C. (1997), The Magma algebra system, I. The userlanguage, J. Symbolic Comput. 24, 235–265.
Bourbaki, N. (1981), Éléments de Mathématiques: Algèbre, Masson.
Bourbaki, N. (1989), Elements of Mathematics: Commutative Algebra, Chapters 1–7, Springer Verlag.
Brauer, A., Brauer, R. and Hopf, H. (1926), Über die Irreduzibilität einiger speziellerKlassen von Polynomen, Jber. Deutsch. Math. Verein. 35, 99–112.
Bremner, A. (1988), On power bases in cyclotomic number fields, J. Number Theor. 28, 288–298.
Brindza, B. (1996), On large values of binary forms, Rocky Mountain J.Math. 26, 839–845.
Brindza, B., Evertse, J.-H. and Győry, K. (1991), Bounds for the solutions of some diophantineequations in terms of discriminants, J. Austral. Math. Soc. Ser.. 51, 8–26.
Brunotte, H. (2001), On trinomial bases of radix representations of algebraic integers, Acta Sci. Math. 67, 521–527.
Brunotte, H., Huszti, A. and Pethő, A. (2006), Bases of canonical number systems inquartic algebraic number fields, J. Theor. Nombres Bordeau. 18, 537–557.
Buchmann, J. and Ford, D. (1989), On the computation of totally real quartic fields ofsmall discriminant, Math. Comp.. 52, 161–174.
Bugeaud, Y. and Dujella, A. (2011), Root separation for irreducible integer polynomials, Bull. London Math. Soc. 43, 1239–1244.
Bugeaud, Y. and Dujella, A. (2014), Root separation for reducible integer polynomials, Acta Arith. 162, 393–403.
Bugeaud, Y. and Mignotte, M. (2004), On the distance between roots of integer polynomials, Proc. Edinb. Math. Soc. 47, 553–556.
Bugeaud, Y. and Mignotte, M. (2010), Polynomial root separation, Intern. J. Number Theor. 6, 587–602.
Cassels, J.W.S. (1959), An Introduction to the Geometry of Numbers, Springer Verlag.
Char, B.W., Geddes, K.O., Gonnet, G.H., Monagan, M.B., Watt, S.M. (eds.) (1988), MAPLE, Reference Manual, Watcom Publications, Waterloo, Canada.
Coates, J. (1969/1970), An effective p-adic analogue of a theorem of Thue III. Thediophantine equation y
2 = x
3 + k
, Acta Arith.. 16, 425–435.
Cohen, H. (1993), A Course in Computational Algebraic Number Theory, Springer Verlag.
Cohen, H. (2000), Advanced Topics in Computational Number Theory, Springer Verlag.
del Corso, I., Dvornicich, R. and Simon, D. (2005), Decomposition of primes innon-maximal orders, Acta Arith. 120, 231–244.
Cougnard, J. (1988), Conditions nécessaires de monogénéité. Applications auxextensions cycliques de degré premier l ≥ 5 d'un corps quadratique imaginaire, J. London Math. Soc. 37, 73–87.
Cougnard, J. and Fleckinger, V. (1990), Modèle de Legendre d'une courbe elliptiqueà multiplication complexe et monogénéité d'anneaux d'entiers II. Acta Arith.
55, 75–81.
Cremona, J. (1999), Reduction of binary cubic and quartic forms, London Math. Soc. ISSN, 1461–1570.
Daberkow, M., Fieker, C., Kluners, J., Pohst, M., Roegner, K. and Wildanger, K. (1997), KANT V4, J. Symbolic Comput.
24, 267–283.
Dedekind, R. (1878), Über die Zusammenhang zwischen der Theorie der Ideale undder Theorie der höheren Kongruenzen, Abh. Konig. Ges. Wissen. Gottinge. 23, 1–23.
Delone, B.N. (Delaunay) (1930), Über die Darstellung der Zahlen durch die binärenkubischen Formen von negativer Diskriminante, Math. Z. 31, 1–26.
Delone, B.N. and Faddeev, D.K. (1940), The theory of irrationalities of the third degree(Russian), Inst. Math. Steklo. 11, Acad. Sci. USSR, Moscow-Leningrad. English translation, Amer. Math. Soc., 1964.
Derksen, H. and Masser, D.W. (2012), Linear equations over multiplicative groups,recurrences, and mixing I, Proc. Lond. Math. Soc. 104, 1045–1083.
Dujella, A. and Pejković, T. (2011), Root separation for reducible monic quartics, Rend. Semin. Mat. Univ. Padov. 126 (2011), 63–72.
Dummit, D.S. and Kisilevsky, H. (1977), Indices in cyclic cubic fields, in: NumberTheory and Algebra, Academic Press, 29–42.
Eisenbud, D. (1994), Commutative Algebra with a View Toward Algebraic Geometry, Springer Verlag.
Evertse, J.-H. (1984a), On equations in S-units and the Thue-Mahler equation, Invent. Math. 75, 561–584.
Evertse, J.-H. (1984b), On sums of S-units and linear recurrences, Compos. Math. 53, 225–244.
Evertse, J.-H. (1992), Reduced bases of lattices over number fields, Indag. Math. N.S. 3, 153–168.
Evertse, J.-H. (1993), Estimates for reduced binary forms, J. Reine Angew. Math. 434, 159–190.
Evertse, J.-H. (1996), An improvement of the quantitative subspace theorem, Compos. Math. 101, 225–311.
Evertse, J.-H. (2004), Distances between the conjugates of an algebraic number, Publ. Math. Debrece. 65, 323–340.
Evertse, J.-H. and Győry, K. (1985), On unit equations and decomposable form equations, J. Reine Angew. Math. 358, 6–19.
Evertse, J.-H. and Győry, K. (1988a), On the number of polynomials and integralelements of given discriminant, Acta. Math. Hung. 51, 341–362.
Evertse, J.-H. and Győry, K. (1988b), Decomposable form equations, in: New Advancesin Transcendence Theory, Proc. conf. Durham 1986, A., Baker, ed., Cambridge University Press pp. 175–202.
Evertse, J.-H. and Győry, K. (1991a), Effective finiteness results for binary forms withgiven discriminant, Compositio Math.. 79, 169–204.
Evertse, J.-H. and Győry, K. (1991b), Thue inequalities with a small number of solutions, in: The Mathematical Heritage of, C. F.Gauss, World Scientific Publ. Comp., pp. 204–224.
Evertse, J.-H. and Győry, K. (1992a), Effective finiteness theorems for decomposableforms of given discriminant, Acta. Arith. 60, 233–277.
Evertse, J.-H. and Győry, K. (1992b), Discriminants of decomposable forms, in: New Trends in Probability. and Statistics., F., Schweiger and, E., Manstavičius (Eds.), Int. Science Publ. pp. 39–56.
Evertse, J.-H. and Győry, K. (1993), Lower bounds for resultants, I, Compositio Math. 88, 1–23.
Evertse, J.-H. and Győry, K. (1997), The number of families of solutions of decomposableform equations, Acta. Arith. 80, 367–394.
Evertse, J.-H. and Győry, K. (2013), Effective results for unit equations over finitelygenerated domains, Math. Proc. Cambridge Phil. Soc. 154, 351–380.
Evertse, J.-H. and Győry, K. (2015), Unit Equations in Diophantine Number Theory, Camb. Stud. Adv. Math. 146, Cambridge University Press.
Evertse, J.-H. and Győry, K. (2016), Effective results for discriminant equations overfinitely generated domains, arXiv 31602.04730.
Evertse, J.-H., Győry, K., Stewart, C.L. and Tijdeman, R. (1988) On S-unit equations intwo unknowns, Invent. math. 92, 461–477.
Evertse, J.-H. and Schlickewei, H.P. (2002), A quantitative version of the Absolute Subspace Theorem, J. Reine Angew. Math. 548, 21–127.
Evertse, J.-H., Schlickewei, H.P. and Schmidt, W.M. (2002), Linear equations invariables which lie in a multiplicative group, Ann. Math. 155, 807–836.
Faltings, G. (1983), Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73, 349–366.
Fincke, V. and Pohst, M. (1983), A procedure for determining algebraic integers ofgiven norm, in: Computer Algebra, Lecture Notes in Computer Sci.. 162, Springer Verlag, 194–202.
Friedman, E. (1989), Analytic formulas for regulators of number fields, Invent. Math. 98, 599–622.
Frohlich, A. and Shepherdson, J.C. (1956), Effective procedures in field theory, Philos. Trans. Roy. Soc. London, Ser.. 248, 407–432.
Fuchs, C., von Kanel, R. and Wustholz, G. (2011), An effective Shafarevich theorem forelliptic curves, Acta Arith. 148, 189–203.
Funakura, T. (1984), On integral bases of pure quartic fields, Math. J. Okayama Univ. 26, 27–41.
Gaal, I. (1986), In homogeneous discriminant form and index form equations and their applications, Publ. Math. Debrece. 33, 1–12.
Gaal, I. (1988), Integral elements with given discriminant over function fields, Acta. Math. Hung. 52, 133–146.
Gaal, I. (2001), Power integral bases in cubic relative extensions, Experimental Math. 10, 133–139.
Gaal, I. (2002), Diophantine equations and power integral bases, Birkhauser.
Gaal, I. and Győry, K. (1999), Index form equations in quintic fields, Acta Arith. 89, 379–396.
Gaal, I. and Nyul, G. (2006), Index form equations in biquadratic fields: the p-adic case, Publ. Math. Debrece. 68, 225–242.
Gaal, I., Pethő, A. and Pohst, M. (1991a), On the resolution of index form equations inbiquadratic number fields, I.
J. Number Theory
38, 18–34.
Gaal, I., Pethő, A. and Pohst, M. (1991b), On the resolution of index form equations inbiquadratic number fields, II.
J. Number Theory
38, 35–51.
Gaal, I., Pethő, A. and Pohst, M. (1991c), On the resolution of index form equations, in: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ACM Press, pp. 185–186.
Gaal, I., Pethő, A. and Pohst, M. (1993), On the resolution of index form equations inquartic number fields, J. Symbolic Computatio. 16, 563–584.
Gaal, I., Pethő, A. and Pohst, M. (1995), On the resolution of index form equations inbiquadratic number fields, III. The bicyclic biquadratic case, J. Number Theor. 53, 100–114.
Gaal, I., Pethő, A. and Pohst, M. (1996), Simultaneous representation of integers by apair of ternary quadratic forms – with an application to index form equations inquartic number fields, J. Number Theor. 57, 90–104.
Gaal, I. and Pohst, M. (2000), On the resolution of index form equations in relativequartic extensions, J. Number Theory. 85, 201–219.
Gaal, I. and Pohst, M. (2002), On the resolution of relative Thue equations, Math. Comput. 71, 429–440.
Gaal, I. and Schulte, N. (1989), Computing all power integral bases of cubic numberfields, Math. Comput. 53, 689–696.
Gan, W.T., Gross, B. and Savin, G. (2002), Fourier coefficients of modular forms on G2, Duke Math. J.
115, 105–169.
Gauss, C.F. (1801), Disquisitiones Arithmeticae (English translation by A.A., Clarke, Yale University Press, 1965).
Gilbert, W.J. (1981), Radix representations of quadratic fields, J. Math. Anal. Appl. 83, 264–274.
Gras, M.N. (1973), Sur les corps cubiques cycliques dont l'anneau des entiers estmonogène, Ann. Sci. Univ. Besancon, Fasc. 6.
Gras, M.N. (1980), Z-bases d'entiers 1, θ, θ
2
, θ
3
dans les extensions cycliques de degré 4 deQ, Publ.Math. Fac. Sci. Besancon, Theorie des Nombres, 1979/1980 et 1980/81.
Gras, M.N. (1983–1984), Non monogénéité de l'anneau des entiers de certaines extensionsabéliennes de Q, Publ. Math. Sci. Besancon, Theorie des Nombres, 1983–1984.
Gras, M.N. (1986), Non monogénéité de l'anneau des entiers des extensions cycliquesde Q de degré premier l ≥ 5, J. Number Theory, 23, 347–353.
Gras, M.N. and Tanoe, F. (1995), Corps biquadratiques monogènes, Manuscripta Math.
86, 63–79.
Grunwald, V. (1885), Intorno all'aritmetica dei sistemi numerici a base negativa conparticolare riguardo al sistema numerico a base negativo-decimale per lo studiodelle sue analogie coll'aritmetica ordinaria (decimale), Giornale di Matematiche di Battaglini. 23, 203–221, 367.
Győry, K. (1972), Sur l'irréductibilité d'une classe des polynômes, II, Publ. Math. Debrece. 19, 293–326.
Győry, K. (1973), Sur les polynômes à coefficients entiers et de discriminant donné, Acta Arith. 23, 419–426.
Győry, K. (1974), Sur les polynômes à coefficients entiers et de discriminant donné II, Publ. Math. Debrece. 21, 125–144.
Győry, K. (1976), Sur les polynômes à coefficients entiers et de discriminant donné III, Publ. Math. Debrece. 23, 141–165.
Győry, K. (1978a), On polynomials with integer coefficients and given discriminant IV, Publ. Math. Debrece. 25, 155–167.
Győry, K. (1978b), On polynomials with integer coefficients and given discriminant V, p-adic generalizations
, Acta Math. Acad. Sci. Hung. 32, 175–190.
Győry, K. (1979), On the number of solutions of linear equations in units of an algebraicnumber field, Comment. Math. Helv. 54, 583–600.
Győry, K. (1979/1980), On the solutions of linear diophantine equations in algebraicintegers of bounded norm, Ann. Univ. Sci. Budapest. Eotvos, Sect. Math. 22–23, 225–233.
Győry, K. (1980a), Explicit upper bounds for the solutions of some diophantine equations, Ann. Acad. Sci. Fenn., Ser A I, Math. 5, 3–12.
Győry, K. (1980b), Résultats effectifs sur la représentation des entiers par des formesdésomposables, Queen's Papers in Pure and Applied Math., No. 56, Kingston, Canada.
Győry, K. (1980c), On certain graphs composed of algebraic integers of a number fieldand their applications I, Publ. Math. Debrece. 27, 229–242.
Győry, K. (1980d), Corps de nombres algébriques d'anneau d'entiers monogènes, Seminaire Delange-Pisot-Poitou (Theorie des nombres), 20e annee, 1978/1979, No. 26, 1–7.
Győry, K. (1981a), On the representation of integers by decomposable forms in severalvariables, Publ. Math. Debrece. 28, 89–98.
Győry, K. (1981b), On S-integral solutions of norm form, discriminant form and indexform equations, Studia Sci. Math. Hung. 16, 149–161.
Győry, K. (1981c), On discriminants and indices of integers of an algebraic numberfield, J. Reine Angew. Math. 324, 114–126.
Győry, K. (1982), On certain graphs associated with an integral domain and their applicationsto Diophantine problems, Publ. Math. Debrece. 29, 79–94.
Győry, K. (1983), Bounds for the solutions of norm form, discriminant form andindex form equations in finitely generated integral domains, Acta Math. Hung. 42, 45–80.
Győry, K. (1984), Effective finiteness theorems for polynomials with given discriminantand integral elements with given discriminant over finitely generated domains, J. Reine Angew. Math. 346, 54–100.
Győry, K. (1992), Upper bounds for the numbers of solutions of unit equations in twounknowns, Lithuanian Math. J. 32, 40–44.
Győry, K. (1994), Upper bounds for the degrees of decomposable forms of given discriminant, Acta. Arith. 66, 261–268.
Győry, K. (1998), Bounds for the solutions of decomposable form equations, Publ.Math. Debrece. 52, 1–31.
Győry, K. (2000), Discriminant form and index form equations, in: Algebraic NumberTheory and Diophantine Analysis
. Walter de Gruyter, pp. 191–214.
Győry, K. (2001), Thue inequalities with a small number of primitive solutions, Periodica Math. Hung. 42, 199–209.
Győry, K. (2006), Polynomials and binary forms with given discriminant, Publ. Math. Debrece. 69, 473–499.
Győry, K. (2008a), On the abc-conjecture in algebraic number fields, Acta Arith. 133, 281–295.
Győry, K. (2008b), On certain arithmetic graphs and their applications to diophantineproblems, Funct. Approx. Comment. Math.. 39, 289–314.
Győry, K. and Papp, Z.Z. (1977), On discriminant form and index form equations, Studia Sci. Math. Hung. 12, 47–60.
Győry, K. and Papp, Z.Z. (1978), Effective estimates for the integer solutions ofnorm form and discriminant form equations, Publ. Math. Debrece. 25, 311–325.
Győry, K., Pink, I. and Pinter, A. (2004), Power values of polynomials and binomialThue-Mahler equations, Publ. Math. Debrece. 65, 341–362.
Győry, K. and Pinter, A. (2008), Polynomial powers and a common generalization ofbinomial Thue-Mahler equations and S-unit equations, in: Diophantine Equations, N., Saradha, Ed. Narosa Publ. House, New Delhi, pp. 103–119.
Győry, K. and Yu, Kunrui (2006), Bounds for the solutions of S-unit equations anddecomposable form equations, Acta Arith. 123, 9–41.
Hall, M. (1937), Indices in cubic fields, Bull. Amer. Math. Soc. 43, 104–108.
Hanrot, G. (1997), Solving Thue equations without the full unit group, Math. Comp. 69, 395–405.
Haristoy, J. (2003), Équations diophantiennes exponentielles, These de docteur, Strasbourg.
Hasse, H. (1980), Number Theory (English translation), Springer Verlag.
Hensel, K. (1908), Theorie der algebraischen Zahlen, Teubner Verlag, Leipzig-Berlin, 1908.
Hermann, G. (1926), Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95, 736–788.
Hermite, C. (1851), Sur l'introduction des variables continues dans la théorie desnombres, J. Reine Angew. Math. 41, 191–216.
Hermite, C. (1854), Sur la théorie des formes quadratiques I, J. Reine Angew. Math. 47, 313–342.
Hermite, C. (1857), Sur le nombre limité d'irrationalités auxquelles se réduisent lesracines des équations à coefficients entriers complexe d'un degré et d'un discriminantdonnés, J. Reine Angew. Math. 53, 182–192.
Hindry, M. and Silverman, J.H. (2000), Diophantine Geometry, An Introduction, Springer Verlag.
Huard, J.G. (1979), Cyclic cubic fields that contain an integer of given index, Lecture Notes in Math. 751, Springer Verlag, Berlin, 195–199.
Humbert, P. (1940), Théorie de la réduction des formes quadratiques définies positivesdans un corps algébrique K fini, Comm. Math. Helv. 12, 263–306.
Humbert, P. (1949), Réduction des formes quadratiques dans un corps algébrique fini, Comm. Math. Helv. 23, 50–63.
Jadrijević, B. (2009a), Establishing the minimal index in a parametric family of bicyclicbiquadratic fields, Periodica Math. Hung. 58, 155–180.
Jadrijević, B. (2009b), Solving index form equations in two parametric families ofbiquadratic fields, Math. Commun. 14, 341–363.
Javanpeykar, A. (2013), Arakelov invariants of Belyi curves, PhD thesis, Universiteit Leiden and l'Universite Paris Sud 11.
Javanpeykar, A. and von Kanel, R. (2014), Szpiro's small points conjecture for cycliccovers, Doc. Math. 19, 1085–1103.
Javanpeykar, A. and Loughran, D. (2015), Good reduction of algebraic groups and flagvarieties, Arch. Math. 104, 133–143.
de Jong, R. and Remond, G. (2011), Conjecture de Shafarevich effective pour les revêtementscycliques, Algebra and Number Theor. 5, 1133–1143.
de Jong, T. (1998), An algorithm for computing the integral closure, J. Symbolic Computatio. 26, 273–277.
Julia, G. (1917), Étude sur les formes binaires non-quadratiques à indéterminées réellesou complexes, Mem. Acad. Sci. l'Inst. Franc. 55, 1–296; see also Julia's Oeuvres, vol. 5.
Jung, H.Y., Koo, J.K. and Shin, D.H. (2014), Application of Weierstrass units to relativepower integral bases, Rev. Mat. Iberoam. 30, 1489–1498.
von Kanel, R. (2011), An effective proof of the hyperelliptic Shafarevich conjecture andapplications, PhD dissertation, ETH Zurich, 54 pp.
von Kanel, R. (2013), On Szpiro's discriminant conjecture, Internat. Math. Res. Notices 1–35. Published online:: doi:10.193/imrn/vnt079.
von Kanel, R. (2014a), An effective proof of the hyperelliptic Shafarevich conjecture, J. Theor. Nombres Bordeau. 26, 507–530.
von Kanel, R. (2014b), Modularity and integral points on moduli schemes. arXiv;1310.7263v2.
Kappe, L.C., and Warren, B. (1989), An elementary test for the Galois group of a quarticpolynomial, Amer. Math. Monthl. 96, 133–137.
Katai, I. and Kovacs, B. (1980), Kanonische Zahlensysteme in der Theorie derquadratischen algebraischen Zahlen, Acta Sci. Math. 42, 99–107.
Katai, I. and Kovacs, B. (1981), Canonical number systems in imaginary quadratic fields, Acta Math. Acad. Sci. Hung. 37, 159–164.
Katai, I. and Szabo, J. (1975), Canonical number systems for complex integers, Acta Sci. Math. 37, 255–260.
Kedlaya, K.S. (2012), A construction of polynomials with squarefree discriminants, Proc. Amer. Math. Soc. 140, 3025–3033.
Kirschenhofer, P. and Thuswaldner, J.M. (2014), Shift radix systems - a survey, Numeration and Substitutio. 2012, 1–59.
Knuth, D.E. (1960), An imaginary number system, Comm. ACM. 3, 245–247.
Knuth, D.E. (1998), The Art of Computer Programming, Vol. 2, Semi-numerical Algorithms, Addison Wesley, 3rd edition.
Korkine, A. and Zolotareff, G. (1873), Sur les formes quadratiques, Math. Ann.
6, 366–389.
Kormendi, S. (1986), Canonical number systems in Q, Acta Sci. Math.
50, 351–357.
Kovacs, B. (1981), Canonical number systems in algebraic number fields, Acta Math. Acad. Sci. Hung. 37, 405–407.
Kovacs, B. (1989), Integral domains with canonical number systems, Publ.Math. Debrece. 36, 153–156.
Kovacs, B. and Pethő, A. (1991), Number systems in integral domains, especially inorders of algebraic number fields, Acta Sci. Math. 55, 287–299.
Kravchenko, R.V., Mazur, M. and Petrenko, B.V. (2012), On the smallest number of generatorsand the probability of generating an algebra, Algebra and Number Theor. 6, 243–291.
Kronecker, L. (1882), Grundzüge einer arithmetischen Theorie der algebraischenGrössen, J. Reine Angew. Math. 92, 1–122.
Lagrange, J.L. (1773), Recherches d'arithmétiques, Nouv. Mem. Acad. Berlin, 265–312; Oeuvres III, 693–758.
Landau, E. (1918), Verallgemeinerung eines Pólyaschen Satzes auf algebraischeZahlkörper, Nachr. Ges. Wiss. Gottingen, 478–488.
Lang, S. (1960), Integral points on curves, Inst. Hautes Etudes Sci. Publ.Math. 6, 27–43.
Lang, S. (1970), Algebraic Number Theory, Addison-Wesley.
Lenstra, A.K., Lenstra, H.W. Jr. and Lovasz, L. (1982), Factoring polynomials withrational coefficients, Math. Ann. 261, 515–534.
Lenstra, H.W. Jr. (2001), Topics in Algebra, Lecture notes. Online available at http://websites.math.leidenuniv.nl/algebra/topics.pdf
Lercier, R. and Ritzenthaler, C. (2012), Hyperelliptic curves and their invariants: geometric,arithmetic and algorithmic aspects, J. Algebr. 372, 595–636.
Levin, A. (2012), Siegel's theorem and the Shafarevich conjecture, J. Theor. Nombres Bordeau. 24, 705–727.
Liang, J. (1976), On the integral basis of the maximal real subfield of a cyclotomic field, J. Reine Angew. Math. 286–287, 223–226.
Liu, Q. (1996), Modèles entiers des courbes hyperelliptiques sur un corps de valuationdiscrète, Trans. Amer. Math. Soc. 348, 4577–4610.
Lockhart, P. (1994), On the discriminant of a hyperelliptic curve, Trans. Amer. Math. Soc. 342, 729–752.
Louboutin, S. (2000), Explicit bounds for residues of Dedekind zeta functions, values of L-functions at s = 1, and relative class numbers, J. Number Theor. 85, 263–282.
Mahler, K. (1933), Zur Approximation algebraischer Zahlen I: Über den grösstenPrimteiler binärer Formen, Math. Ann. 107, 691–730.
Mahler, K. (1937), Über die Annäherung algebraischer Zahlen durch periodische Algorithmen, Acta Math. 68, 109–144.
Mahler, K. (1964a), Inequalities for ideal bases in algebraic number fields, J. Austral. Math. Soc. 4, 425–428.
Mahler, K. (1964b), An inequality for the discriminant of a polynomial, Michigan Math. J.
11, 257–262.
Markoff, A. (1879), Sur les formes quadratiques binaires indéfinies, Math. Ann. 15, 381–406.
Mason, R.C. (1983), The hyperelliptic equation over function fields, Math. Proc. Camb. Phil. Soc. 99, 219–230.
Mason, R.C. (1984), Diophantine equations over function fields, Cambridge University Press.
Matsumoto, R. (2000), On computing the integral closure, Comm. Algebr. 28, 401–405.
Matsumura, H. (1986), Commutative Ring Theory, Cambridge University Press.
Matveev, E.M. (2000), An explicit lower bound for a homogeneous rational linear formin logarithms of algebraic numbers, II. Izvestiya: Mathematics
64, 1217–1269.
McFeat, R.B. (1971), Geometry of numbers in adéle spaces, Dissertationes Math. 88, Warszawa.
Merriman, J.R. and Smart, N.P. (1993a), The calculation of all algebraic integers ofdegree 3 with discriminant a product of powers of 2 and 3 only, Publ.Math. Debrece. 43, 105–205.
Merriman, J.R. and Smart, N.P. (1993b),
Curves of genus 2 with good reduction awayfrom 2 with a rational Weierstrass point
, Math. Proc. Cambridge Philos. Soc.
114, 203–214. Corrigenda: ibid. 118 (1995), 189.
Mignotte, M. and Payafar, M. (1978), Distance entre les racines d'un polynôme, RAIRO Anal. Numer. 13, 181–192.
Mordell, L.J. (1945), On numbers represented by binary cubic forms, Proc. London Math. Soc. 48, 198–228.
Mordell, L.J. (1969), Diophantine Equations, Academic Press, New York, London.
Motoda, Y. and Nakahara, T. (2004),
Power integral bases in algebraic number fieldswhose Galois groups are 2-elementary abelian
, Arch. Math. 83, 309–316.
Nagata, M. (1956), A general theory of algebraic geometry over Dedekind domains I, Amer. J. Math. 78, 78–116.
Nagell, T. (1930), Zur Theorie der kubischen Irrationalitäten, Acta Math. 55, 33–65.
Nagell, T. (1965), Contributions à la théorie des modules et des anneaux algébriques, Arkiv for Mat.. 6, 161–178.
Nagell, T. (1967), Sur les discriminants des nombres algébriques, Arkiv for Mat. 7, 265–282.
Nagell, T. (1968), Quelques propriétés des nombres algébriques du quatrième degré, Arkiv for Mat. 7, 517–525.
Nakagawa, J. (1989), Binary forms and orders of algebraic number fields, Invent. Math. 97, 219–235.
Nakahara, T. (1982), On cyclic biquadratic fields related to a problem of Hasse, Monatsh. Math. 94, 125–132.
Nakahara, T. (1983), On the indices and integral bases of non-cyclic but abelianbiquadratic fields, Archiv der Math. 41, 504–508.
Nakahara, T. (1987), On the minimum index of a cyclic quartic field, Archiv derMath. 48, 322–325.
Narkiewicz, W. (1974), Elementary and analytic theory of algebraic numbers, Springer Verlag/PWN-Polish Scientific Publishers; 2nd ed. (1990).
Neukirch, J. (1999), Algebraic Number Theory, transl. from German by, N., Schappacher, Springer Verlag.
Nguyen, K.D. (2014), On modules of integral elements over finitely generated domains, arXiv:1412.2868v3, Trans. Amer. Math. Math. Soc.
O'Leary, R. and Vaaler, J.D. (1993), Small solutions to inhomogeneous linear equationsover number fields, Trans. Amer. Math. Soc. 326, 915–931.
Oort, F. (1974), Hyperelliptic curves over number fields, in: Classification of Algebraic Varieties and Compact Complex Manifolds, Lecture Notes Math. 412, Springer Verlag, pp. 211–218.
Parry, C.J. (1950),
The p-adic generalization of the Thue-Siegel theorem
, Acta Math. 83, 1–100.
Parshin, A.N. (1968), Algebraic curves over function fields I, Izv. Akad. Nauk. SSSR Ser. Mat.
32, 1191–1219; English transl. in Math. USSR Izv. 2, 1145–1170.
Parshin, A.N. (1972), Minimal models of curves of genus 2 and homomorphisms of abelian varieties defined over a field of finite characteristic, Izv. Akad. Nauk. SSSR Ser. Mat.
36, 67–109; English transl. in Math. USSR Izv. 6, 65–108.
Penney, W. (1965), A “binary” system for complex numbers, J. ACM. 12, 247–248.
Peruginelli, G. (2014), Integral-valued polynomials over the set of algebraic integers of bounded degree, J. Number Theor. 137, 241–255.
Pethő, A. (1991), On a polynomial transformation and its application to the constructionof a public key cryptosystem, in:, A., Pethő, M., Pohst, H.G., Zimmer and H.C., Williams (eds.), pp. 31–44.
Pethő, A. (2004), Connections between power integral bases and radix representationsin algebraic number fields, in: Yokoi-Chowla Conjecture and Related Problems, Furukawa Total Printing Co. LTD, Saga, Japan. pp. 115–125.
Pethő, A. and Pohst, M. (2012), On the indices of multiquadratic number fields, Acta Arith. 153, 393–414.
Pethő, A. and Schulenberg, R. (1987), Effektives Lösen von Thue Gleichungen, Publ. Math. Debrece. 34, 189–196.
Pethő, A. and Ziegler, V. (2011), On biquadratic fields that admit unit power integralbasis, Acta Math. Hung. 133, 221–241.
Petsche, C. (2012), Crirically separable rational maps in families, Compositio Math. 148, 1880–1896.
Pinter, A. (1995), On the magnitude of integer points on elliptic curves, Bull. Austral. Math. Soc. 52, 195–199.
Pleasants, P.A.B. (1974), The number of generators of the integers of a number field, Mathematik. 21, 160–167.
Pohst, M.E. (1982), On the computation of number fields of small discriminants includingthe minimum discriminant of sixth degree fields, Math. Proc. Camb. Philos. Soc.
114, 203–214.
Pohst, M.E. (1993), Computational Algebraic Number Theory, Birkhauser.
Pohst, M.E. and Zassenhaus, H. (1989), Algorithmic algebraic number theory, Cambridge University Press.
van der Poorten, A.J. and Schlickewei, H.P. (1982), The growth condition for recurrencesequences, Macquarie Univ. Math. Rep. 82–0041, North Ryde, Australia.
Rabin, M.O. (1960), Computable Algebra, General Theory and Theory of ComputableFields, Trans. Am.Math. Soc. 95, 341–360.
Ranieri, G. (2010), Power bases for rings of integers of abelian imaginary fields, J. London Math. Soc. (2). 82, 144–160.
Ribenboim, P. (2001), Classical theory of algebraic numbers, Springer Verlag.
Ribenboim, P. (2006), Finite sets of binary forms, Publ. Math. Debrecen. 68, 261–282.
Roberts, D.P. (2015), Polynomials with prescribed bad primes, Intern. J. Number Theor. 11, 1115–1148.
Robertson, L. (1998), Power bases for cyclotomic integer rings, J. Number Theor. 69, 98–118.
Robertson, L. (2001),
Power bases for 2-power cyclotomic fields
, J. Number Theor. 88, 196–209.
Robertson, L. (2010), Monogeneity in cyclotomic fields, Int. J. Number Theor. 6, 1589–1607.
Robertson, L. and Russel, R. (2015), A hybrid Gröbner bases approach to computingpower integral bases, Acta Math. Hung. 147, 427–437.
Roquette, P. (1957), Einheiten und Divisorenklassen in endlich erzeugbaren Körpern, J be. Deutsch. Math. Verei. 60, 1–21.
Rosser, J.B. and Schoenfeld, L. (1962), Approximate formulas for some functions ofprime numbers, Illinois J. Math. 6, 64–94.
Schertz, R. (1989), Konstruktion von Potenzganzheitsbasen in Strahlklassenkörpernüber imaginär-quadratischen Zahlkörpern, J. Reine Angew. Math. 398, 105–129.
Schlickewei, H.P. (1977),
The p-adic Thue-Siegel-Roth-Schmidt theorem
, Arch. Math. (Basel). 29, 267–270.
Schmidt, W.M. (1972), Norm form equations, Ann. Math. 96, 526–551.
Schmidt, W.M. (1980), Diophantine approximation, Lecture Notes Math. 785, Springer Verlag.
Schmidt, W.M. (1991), Diophantine approximations and diophantine equations, Lecture Notes Math. 1467, Springer Verlag.
Schmidt, W.M. (1996), Heights of points on subvarieties of G nm, in: Number Theory 1993–94, London Math. Soc. Lecture Note Ser. 235, S., David, ed., Cambridge University Press. 157–187.
Schonhage, A. (2006), Polynomial root separation examples, J. Symbolic Comput. 41 (2006), 1080–1090.
Schulte, N. (1989), Indexgleichungen in kubischen Zahlkörpern, Diplomarbeit, Dusseldorf.
Schulte, N. (1991), Index form equations in cubic number fields, in: Computational Number Theory, de Gruyter, pp. 281–287.
Seidenberg, A. (1974), Constructions in algebra, Trans. Amer. Math. Soc. 197, 273–313.
Serra, M. (2013), Smooth models of curves, Master thesis, Erasmus Mundus Algant, Universiteit Leiden.
Shafarevich, I.R. (1963), Algebraic number fields, in: Proc. Intern. Congr.Math. (Stockholm, 1962), Inst. Mittag-Leffler,Djursholm, pp. 163–176; English transl. in Amer. Math. Soc. Trans. 31 (1963), 25–39.
Shlapentokh, A. (1996), Polynomials with a given discriminant over fields of algebraicfunctions of positive characteristic, Pacific J. Math. 173, 533–555.
Siegel, C.L. (1921), Approximation algebraischer Zahlen, Math., Z. 10, 173–213.
Silverman, J.H. (2009), The arithmetic of elliptic curves, 2nd ed., Springer Verlag.
Simmons, H. (1970), The solution of a decision problem for several classes of rings, Pacific J. Math. 34, 547–557.
Simon, D. (2001), The index of nonmonic polynomials, Indag. Math. (N.S). 12, 505–517.
Simon, D. (2003), La classe invariante d'une forme binaire, C.R. Math. Acad. Sci. Pari. 336, 7–10.
Smart, N. (1993), Solving a quartic discriminant form equation, Publ. Math. Debrece. 43, 29–39.
Smart, N. (1995), The solution of triangularly connected decomposable form equations, Math. Comp. 64, 819–840.
Smart, N.P. (1996), Solving discriminant form equations via unit equations, J. Symbolic Comp. 21, 367–374.
Smart, N.P. (1997), S-unit equations, binary forms and curves of genus 2, Proc. London Math. Soc. (3) 75, 271–307.
Smart, N.P. (1998), The Algorithmic Resolution of Diophantine Equations, Cambridge University Press.
Spearman, B.K. and Williams, K.S. (2001), Cubic fields with a power basis, Rocky Mountain J. Math. 31, 1103–1109.
Stark, H.M. (1974), Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23, 135–152.
Stewart, C.L. (1991), On the number of solutions of polynomial congruences and Thueequations, J. Amer.Math. Soc. 4, 793–835.
Stothers, W.W. (1981), Polynomial identities and Hauptmoduln, Quart. J.Math. Oxford Ser. (2. 32, 349–370.
Stout, B.J. (2014), A dynamical Shafarevich theorem for twists of rational morphisms, Acta Arith. 166, 69–80.
Szpiro, L. and Tucker, T.J. (2008), A Shafarevich-Faltings theorem for rational functions, Pure Appl. Math. Quartel. 4, 1–14. The PARI Group (2004), Bordeaux, PARI/GP, version 2.1.5, available from http://pari.math.u-bordeaux.fr/.
Therond, J.D. (1995), Extensions cycliques cubiques monogènes de l'anneau des entiersd'un corps quadratique, Archiv der Math. 64, 216–229.
Thunder, J.L. (1995), On Thue ineqalities and a conjecture of Schmidt, J. Number Theor. 52, 319–328.
Thunder, J.L. and Wolfskill, J. (1996), Algebraic integers of small discriminant, Acta Arith. 75, 375–382.
Trelina, L.A. (1977a), On algebraic integers with discriminants having fixed primedivisors, Mat. Zametk. 21, 289–296 (Russian).
Trelina, L.A. (1977b), On the greatest prime factor of an index form, Dokl. Akad. Nauk BSS. 21, 975–976 (Russian).
Trelina, L.A. (1985), Representation of powers by polynomials in algebraic numberfields, Dokl. Akad. Nauk BSSR. 29, 5–8 (Russian).
Tzanakis, N. and de Weger, B.M.M. (1989), On the practical solution of the Thue equation, J. Number Theor. 31, 99–132.
Vaaler, J.D. (2014), Heights on groups and small multiplicative dependencies, Trans. Amer. Math. Soc. 366, 3295–3323.
Voloch, J.F. (1998), The equation ax + by = 1 in characteristic p
, J. Number Theor. 73, 195–200.
van der Waerden, B.L. (1930), Moderne Algebra I (1st ed.), Julius Springer.
Waldschmidt, M. (2000), Diophantine approximation on linear algebraic groups, Springer Verlag.
Wildanger, K. (1997), Über das Lösen von Einheiten- und Index formgleichungen inalgebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzenPunkte einer Mordellschen Kurve, Dissertation, Technical University, Berlin.
Wildanger, K. (2000), Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern, J. Number Theor. 82, 188–224.
Wood, M.M. (2011), Rings and ideals parameterized by binary n-ic forms, J. London Math. Soc. 83, 208–231.
Yingst, A.Q. (2006), A characterization of homeomorphic Bernoulli trialmeasures, PhD dissertation, Univ. North Texas.
Yu, K. (2007), P-adic logarithmic forms and group varieties III, Forum Mathematicu. 19, 187–280.
Zhuang, W. (2015), Symmetric Diophantine approximation over function fields, PhD dissertation, Universiteit Leiden.
