Research - Kálmán Győry

My main research interests are in Number Theory, in particular in Diophantine Number Theory, including Diophantine equations and inequalities over Z, over (rings of inegers or S-integers of) number fields, function fields and even more generally, over finitely generated domains which may contain transcendental elements, too. My results are mainly related to unit equations, decomposable form equations, discriminant and index equations and power integral bases which are of central importance in Number Theory.  The results have numerous consequences and applications in Diophantine Number Theory, Algebraic Number Theory and Algoritmic Number Theory. Below there is a brief selection from my scientific achievements and my publications. The selection of results was in fact made by Professor Á. Pintér in his talk given at the "Győry 75 Symposium" in Debrecen in 2015.

Effective results over Z and over number fields

  • The first explicit bounds for the solutions of unit and S-unit equations over number fields (Gy); improvements of the bounds (partly with Bugeaud and Yu).
  • Extensive applications of the bounds mentioned among others to discriminant and index equations, power integral bases, irreducible polynomials, arithmetic graphs (Gy) and decomposable form equations (partly with Papp, Evertse, Bugeaud and Yu).
  • The first effective bounds for the solutions of discriminant form and index form equations (Gy); generalizations (partly with Papp and Evertse).
  • Generalization of Baker’s quantitative effective theorem on Thue equations to a large class of decomposable form equations in an arbitrary number of unknowns (with Papp); several improvements and further generilaziations (partly with Evertse, Bugeaud and Yu).
  • Common generalization of effective finiteness results on S-unit equations and binomial Thue equations with unknown exponents (with Pink and Pintér).
  • The upper bound Card(S) +1 for the number of solutions of S-unit equations in a given number field, apart from finitely many and effectively determinable S-equivalence classes of equations (with Evertse, Stewart and Tijdeman).
  • The first effective finiteness theorems for monic polynomials, algebraic integers (Gy), binary forms and decomposable forms (with Evertse) with given nonzero discriminant; improvements, generalizations, bounds for the degrees and number of solutions and various applications (partly with Evertse).
  • New information in effective and quantitative form on the arithmetical properties of discriminants of monic polynomials, algebraic integers (Gy), binary forms and decomposable forms (with Evertse); improvements, generalizations and applications (partly with Evertse).
  • The first general algorithm for deciding the monogeneity and multiple monogeneity of number fields, and for determining all power integral bases (Gy); generalizations among others to orders in number fields (Gy) and in étale algebras over number fields (with Evertse); applications to generalized number systems in étale orders (with Evertse, Pethő and Thuswaldner).
  • The results mentioned above provided the solutions of several old problems and confirmed some conjectures.

Ineffective results over finitely generated domains over Z

  • Finiteness theorems for monic polynomials and integral elements with given degree and given non-zero discriminant, and for power integral bases (Gy, partly with Evertse) for multiply monogenic orders over finitely generated domains (with Bérczes and Evertse); uniform bounds for the numbers of solutions (with Evertse).
  • Extending the finiteness theorems of Evertse and van der Poorten-Schlickewei concerning the number of non-degenerate solutions of multivariate unit equations over number fields K to the case of finitely generated fields K over Q, with solutions from a finitely generated multiplicative subgroup  Γ of K* (with Evertse); the first uniform bound, independent of the coefficients, for the number of such solutions (with Evertse).
  • In case of two unknowns, the sharp upper bound 2 for the number of solutions from Γ, apart from finitely many Γ -equivalence classes of equations over K (with Evertse, Stewart and Tijdeman). Several applications, e.g. to Thue-Mahler equations (with Evertse).
  • Generalization of finiteness criteria of Schmidt, Schlickewei and Laurent concerning the solutions of norm equations to arbitrary decomposable form equations, including discriminant form equations and index  form equations (with Evertse).
  • Generalization of finiteness theorems of Schmidt, Schlickewei and Laurent concerning the families of solutions of norm form equations to arbitrary decomposable form equations (Gy); description of the structure of the set of solutions of systems of decomposable form equations (Gy); quantitative versions in the number fields case, uniform bounds for the number of families of solutions, as well as for the number of solutions when this number is finite (with Evertse); applications to the number of prime factors of integers of the form  a+b   resp. ab+1 and  to their generalizations (with Stewart and Tijdeman  resp,  Sárközy and Stewart); density results in the case when the number of solutions is infinite (with Pethő and Everest).

Effective results and methods over finitely generated domains over Z

  • Effective specialization method for extending the effective theory of diophantine equations over number fields to the finitely generated case (Gy, and a refined and extended form with Evertse).
  • General effective finiteness results in quantitative form for unit equations, decomposable form equations and discriminant equations (with Evertse), and for Thue equations, hyper- and superelliptic equations and the Schinzel-Tijdeman equation (with Bérczes and Evertse) over finitely generated domains.

Miscellaneous explicit results

  • General irreducibility theorems for polynomials of the form  g(f(X)) and for neighbouring polynomials over  Z (Gy).
  • Norm inequalities and discriiminant inequalities in CM-fields and their applications (Gy).
  • Characterizations of certain arithmetic graphs associated with integral domains and their applications to diophantine problems (Gy, and Gy partly with Hajdu and Tijdeman).
  • Complete solution of some infinite classes of binomial Thue equations and ternary equations with unknown exponents and unknown S-unit coefficients, extension of Wiles’ theorem concerning the Fermat equation to more general ternary equations (with Bennett, Mignotte and Pintér).
  • As an extension of Erdős, Darmon and Merel, finding all binomial coefficients which are perfect powers (Gy).
  • Finding all perfect powers in sums of consecutive k-th powers (with Benett and Pintér), and in products of consecutive terms of arithmetic progression (with Benett, Bruin, Hajdu, Pintér and Saradha).

Main publications

Books

  1. K. Győry, Résultats effectifs sur la représentation des entiers par des formes décomposables. Queen's Papers in Pure and Applied Math., No. 56, Kingston, Canada, 1980.
  2. J.-H. Evertse and K. Győry, Unit Equations in Diophantine Number Theory, Cambridge University Press, 2015.
  3. J.-H. Evertse and K. Győry, Discriminant Equations in Diophantine Number Theory, Cambridge University Press, 2016.
  4. J.-H. Evertse and K.Győry, Effective results and methods for Diophantine equations over finitely generated domains, Cambridge University Press, 2022.

The books provided the first comprehensive treatments of their subjects. Several new results and applications were included in the books.

Survey papers

  1. K. Győry, Corps de nombres algébriques d'anneau d'entiers monogéne. Séminaire Delange-Pisot-Poitou (Théorie des Nombres), 20e année, 1978/1979. Paris, No 26, 1-7 (1980).
  2. J.-H. Evertse, K. Győry, C. L. Stewart and R. Tijdeman, S-unit equations and their applications. New Advances in Transcendence Theory (A. Baker ed.), pp. 110-174. Cambridge University Press, 1988.
  3. J.-H. Evertse and K. Győry, Decomposable form equations. New Advances in Transcendence Theory (A. Baker ed.), pp. 175-202. Cambridge University Press, 1988.
  4. K. Győry, Some recent applications of S-unit equations. Astérisque, 209. Soc. Math. France,1992. pp. 17-38.
  5. K. Győry, On the distribution of solutions of decomposable form equations. Number Theory in Progress. Walter de Gruyter, Berlin-New York, 1999. pp. 237-265.
  6. K. Győry, Discriminant form and index form equations. Algebraic Number Theory and Diophantine Analysis. Walter de Gruyter, Berlin-New York, 2000. pp. 191-214.
  7. K. Győry, Solving diophantine equations by Baker's theory. A Panorama of Number Theory, Cambridge University Press, 2002. pp. 38-72.
  8. K. Győry, Polynomials and binary forms with given discriminant, Publ. Math. Debrecen, 69 (2006), 473-499.
  9. K. Győry, Perfect powers in products with consecutive terms from arithmetic progressions II, in: Erdős Centennial, Bolyai Soc. Math. Stud., Springer, 2013, 311-324.
  10. J.-H. Evertse, K. Győry, Effective results for Diophantine equations over finitely generated domains: A survey, in: Turán memorial, Number Theory, Analysis and Combinatorics: Proceedings of the Paul Turan memorial conference, Budapest: De Gruyter, (2015) 63-74.

Research papers

  1. K.Győry, Sur l'irréductibilité d'une classe des polynômes I, Publ. Math.Debrecen 18 (1971), 289-307; Part II, ibid, 19 (1972), 293-326; On the irreducibility of a class of polynomials III, J.Number Theory 15 (1982), 164-181; Part IV, Acta Arith. 62 (1992), 399-405.
  2. K.Győry, Sur les polynômes à coefficients entiers et de discriminant donné, Acta Arith. 23 (1973), 419-426 ; Part II, Publ.Math.Debrecen 21 (1974), 125-144; Part III, ibid. 23 (1976), 121-165; On polynomials with integer coefficients and given discriminant IV, ibid. 25 (1978),155-167; Part V, P-adic generalizations, Acta Math.Acad. Sci.Hungar. 32 (1978), 175-190.
  3. K. Győry, Sur une classe des corps de nombres algébriques et ses applications. Publ. Math. Debrecen 22 (1975), 151-175.
  4. K. Győry and Z. Z. Papp, Effective estimates for the integer solutions of norm form and discriminant form equations. Publ. Math. Debrecen 25 (1978), 311-325.
  5. M. Voorhoeve, K. Győry and R. Tijdeman, On the Diophantine equation $1\sp+2\sp+\cdots +x\sp+R(x)=y\sp$. Acta Math. 143 (1979), 1-8.
  6. K. Győry, On the number of solutions of linear equations in units of an algebraic number field. Comment. Math. Helv. 54 (1979), 583-600.
  7. K. Győry, On discriminants and indices of integers of an algebraic number field. J. Reine Angew. Math. 324 (1981), 114-126.
  8. K. Győry, On certain graphs associated with an integral domain and their applications to diophantine problems. Publ. Math. Debrecen 29 (1982), 79-94.
  9. K. Győry, Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains. Acta Math. Hungar. 42 (1983), 45-80.
  10. K. Győry, Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely generated domains. J. Reine Angew. Math. 346 (1984), 54-100.
  11. J.-H. Evertse and K. Győry, On unit equations and decomposable form equations. J. Reine Angew. Math. 358 (1985), 6-19.
  12. K. Győry, C. L. Stewart and R. Tijdeman, On prime factors of sums of integers I. Compositio Math. 59 (1986), 81-89.
  13. J.-H. Evertse, K. Győry, C. L. Stewart and R. Tijdeman, On S-unit equations in two unknowns. Invent. Math. 92 (1988), 461-477.
  14. J.-H. Evertse and K. Győry, On the numbers of solutions of weighted unit equations. Compositio Math. 66 (1988), 329-354.
  15. J.-H. Evertse and K. Győry, Finiteness criteria for decomposable form equations. Acta Arith. 50 (1988), 357-379.
  16. J.-H. Evertse and K. Győry, Thue-Mahler equations with a small number of solutions. J. Reine Angew. Math. 399 (1989), 60-80.
  17. J.-H. Evertse and K. Győry, Effective finiteness results for binary forms with given discriminant. Compositio Math. 79 (1991), 169-204.
  18. J.-H. Evertse and K. Győry, Effective finiteness theorems for decomposable forms of given discriminant. Acta Arith. 60 (1992), 233-277.
  19. K. Győry, On the numbers of families of solutions of systems of decomposable form equations. Publ. Math. Debrecen 42 (1993), 65-101.
  20. K. Győry, On the irreducibility of neighbouring polynomials. Acta Arith. 67 (1994), 283-294.
  21. Y. Bugeaud and K. Győry, Bounds for the solutions of unit equations. Acta Arith. 74 (1996), 67-80.
  22. Y. Bugeaud and K. Győry, Bounds for the solutions of Thue-Mahler equations and norm form equations. Acta Arith. 74 (1996), 273-292.
  23. K. Győry, A. Sárközy and C. L. Stewart, On the number of prime factors of integers of the form ab+1. Acta Arith. 74 (1996), 365-385.
  24. K. Győry, On the diophantine equation /--binomial coefficient  n  over  k = x  to the  l--/ Acta Arith. 80 (1997), 289-295.
  25. J.-H. Evertse and K. Győry, The number of families of solutions of decomposable form equations. Acta Arith. 80 (1997), 367-394.
  26. I. Gaál and K. Győry, Index form equations in quintic fields. Acta Arith. 89 (1999), 379-396.
  27. M. Bennett, K. Győry and Á. Pintér, On the diophantine equation /--1 to the k + ... +  x to the k = y to the  n--/. Compositio Math., 140 (2004), 1417-1431.
  28. K. Győry, I. Pink and Á. Pintér, Power values of polynomials and binomial Thue-Mahler equations. Publ. Math. Debrecen, 65 (2004), 341-362.
  29. M. Bennett, N. Bruin, K. Győry and L. Hajdu, Powers from products of consecutive terms in arithmetic progressions. Proc. London Math. Soc., (3) 92 (2006), 273-306.
  30. M. Bennett, K. Győry, M. Mignotte and Á. Pintér, Binomial Thue equations and polynomial powers. Compositio Math., 142 (2006), 1103-1121.
  31. K. Győry and K. Yu, Bounds for the solutions of S-unit equations and decomposable form equations, Acta Arith., 123 (2006), 9-41.
  32. K. Győry, L. Hajdu and Á. Pintér, Perfect powers from products of consecutive terms in arithmetic progression, Compositio Math., 145 (2009), 845-864.
  33. A. Bérczes, J.-H. Evertse and K. Győry, Multiply monogenic orders, Ann. Scuola Normale Sup. Pisa Cl. Sci. (5) 12 (2013), 467-497.
  34. J.-H. Evertse and K. Győry, Effective results for unit equations over finitely generated domains, Math. Proc Cambridge Philos. Soc. 154 (2013), 351-380.
  35. A. Bérczes, J.-H. Evertse and K. Győry, Effective results for Diophantine equations over finitely generated domains, Acta Arith., 163 (2014), 71-100.
  36. J.-H. Evertse, K. Győry, A. Pethő and J. Thuswaldner, Number systems over general orders, Acta Math.Hungar, 159 (2019), 187-205.
  37. K. Győry, Bounds for the solutions of S-unit equations and decomposable form equations II., Publ. Math. Debrecen, 94 (2019), 507-526. Corregindum: ibid. 97 (2020), 525

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Last update: 2023. 07. 05. 12:53