Effective Results for Discriminant Equations over Finitely Generated Integral Domains

Abstract

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 F ∈ A[X]. Further, given a finite étale K-algebra $\Omega$, denote by $D_{\Omega /K}(\alpha )$ the discriminant of α over K. For non-zero δ ∈ A, we consider equations

$$\displaystyle{D(F) =\delta }$$

to be solved in monic polynomials F ∈ A[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.

To Professor Robert Tichy on the occasion of his 60th birthday