Effective Results for Discriminant Equations over Finitely Generated Integral Domains



Let A be an integral domain with quotient field K of characteristic 0 that is finitely generated as a \(\mathbb{Z}\)-algebra. Denote by D(F) the discriminant of a polynomial FA[X]. Further, given a finite étale K-algebra \(\Omega\), denote by \(D_{\Omega /K}(\alpha )\) the discriminant of α over K. For non-zero δA, we consider equations
$$\displaystyle{D(F) =\delta }$$
to be solved in monic polynomials FA[X] of given degree n ≥ 2 having their zeros in a given finite extension field G of K, and
$$\displaystyle{D_{\Omega /K}(\alpha ) =\delta \,\, \mbox{ in }\alpha \in O,}$$
where O is an A-order of \(\Omega\), i.e., a subring of the integral closure of A in \(\Omega\) that contains A as well as a K-basis of \(\Omega\).

In the series of papers (Győry, Acta Arith 23:419–426, 1973; Győry, Publ Math Debrecen 21:125–144, 1974; Győry, Publ Math Debrecen 23:141–165, 1976; Győry, Publ Math Debrecen 25:155–167, 1978; Győry, Acta Math Acad Sci Hung 32:175–190, 1978; Győry, J Reine Angew Math 324:114–126, 1981), Győry proved that when K is a number field, A the ring of integers or S-integers of K, and \(\Omega\) a finite field extension of K, then up to natural notions of equivalence the above equations have, without fixing G, finitely many solutions, and that moreover, if K, S, \(\Omega\), O, and δ are effectively given, a full system of representatives for the equivalence classes can be effectively determined. Later, Győry (Publ Math Debrecen 29:79–94, 1982) generalized in an ineffective way the above-mentioned finiteness results to the case when A is an integrally closed integral domain with quotient field K of characteristic 0 which is finitely generated as a \(\mathbb{Z}\)-algebra and G is a finite extension of K. Further, in Győry (J Reine Angew Math 346:54–100, 1984) he made these results effective for a special class of integral domains A containing transcendental elements. In Evertse and Győry (Discriminant equations in diophantine number theory, Chap.  10 Cambridge University Press, 2016) we generalized in an effective form the results of Győry (Publ Math Debrecen 29:79–94, 1982) mentioned above to the case where A is an arbitrary integrally closed domain of characteristic 0 which is finitely generated as a \(\mathbb{Z}\)-algebra, where \(\Omega\) is a finite étale K-algebra, and where A, δ, and G, respectively \(\Omega,O\) are effectively given (in a well-defined sense described below).

In the present paper, we extend these effective results further to integral domains A that are not necessarily integrally closed.

2010 Mathematics Subject Classification:

11D99 Secondary 11D41 



We would like to thank the two anonymous referees for their careful scrutiny of our paper and their valuable comments and corrections.


  1. 1.
    M. Aschenbrenner, Ideal membership in polynomial rings over the integers. J. Am. Math. Soc. 17, 407–442 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    I. Borosh, M. Flahive, D. Rubin, B. Treybig, A sharp bound for solutions of linear diophantine equations. Proc. Am. Math. Soc. 105, 844–846 (1989)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry (Springer, Berlin, 1994)MATHGoogle Scholar
  4. 4.
    J.-H. Evertse, K. Győry, Effective results for unit equations over finitely generated domains. Math. Proc. Camb. Philos. Soc. 154, 351–380 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    J.-H. Evertse, K. Győry, Discriminant Equations in Diophantine Number Theory (Cambridge University Press, 2016)Google Scholar
  6. 6.
    K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné. Acta Arith. 23, 419–426 (1973)MathSciNetMATHGoogle Scholar
  7. 7.
    K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné II. Publ. Math. Debrecen 21, 125–144 (1974)MathSciNetMATHGoogle Scholar
  8. 8.
    K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné III. Publ. Math. Debrecen 23, 141–165 (1976)MathSciNetMATHGoogle Scholar
  9. 9.
    K. Győry, On polynomials with integer coefficients and given discriminant IV. Publ. Math. Debrecen 25, 155–167 (1978)MathSciNetMATHGoogle Scholar
  10. 10.
    K. Győry, On polynomials with integer coefficients and given discriminant V, \(\mathfrak{p}\)-adic generalizations. Acta Math. Acad. Sci. Hung. 32, 175–190 (1978)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    K. Győry, On discriminants and indices of integers of an algebraic number field. J. Reine Angew. Math. 324, 114–126 (1981)MathSciNetMATHGoogle Scholar
  12. 12.
    K. Győry, On certain graphs associated with an integral domain and their applications to diophantine problems. Publ. Math. Debrecen 29, 79–94 (1982)MathSciNetMATHGoogle Scholar
  13. 13.
    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, 54–100 (1984)MathSciNetMATHGoogle Scholar
  14. 14.
    G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95, 736–788 (1926)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    T. de Jong, An algorithm for computing the integral closure. J. Symb. Comput. 26, 273–277 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    R. Matsumoto, On computing the integral closure. Commun. Algebra 28, 401–405 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    H. Matsumura, Commutative Ring Theory (Cambridge University Press, Cambridge, 1986)MATHGoogle Scholar
  18. 18.
    M. Nagata, A general theory of algebraic geometry over Dedekind domains I. Am. J. Math. 78, 78–116 (1956)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    P. Roquette, Einheiten und Divisorenklassen in endlich erzeugbaren Körpern. Jahresber. Deutsch. Math. Verein 60, 1–21 (1957)MathSciNetMATHGoogle Scholar
  20. 20.
    A. Seidenberg, Constructions in algebra. Trans. Am. Math. Soc. 197, 273–313 (1974)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Mathematical InstituteLeiden UniversityLeidenThe Netherlands
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecen, Egyetem Tér 1Hungary

Personalised recommendations