Effective results for discriminant equations over finitely generated 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 $F\in A[X]$. Further, given a finite etale algebra $\Omega$, we denote by $D_{\Omega/K}(\alpha )$ the discriminant of $\alpha$ over $K$. For non-zero $\delta\in A$, we consider equations \[ D(F)=\delta \] to be solved in monic polynomials $F\in A[X]$ of given degree $n\geq 2$ having their zeros in a given finite extension field $G$ of $K$, and \[ D_{\Omega/K}(\alpha)=\delta\,\,\mbox{ in } \alpha\in O, \] 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 our book ``Discriminant Equations in Diophantine Number Theory, which will be published by Cambridge University Press we proved that if $A$ is effectively given in a well-defined sense and integrally closed, then up to natural notions of equivalence the above equations have only finitely many solutions, and that moreover, a full system of representatives for the equivalence classes can be determined effectively. In the present paper, we extend these results to integral domains $A$ that are not necessarily integrally closed.