Effective results for discriminant equations over finitely generated domains

TL;DR

This paper extends effective finiteness results for discriminant equations over finitely generated integral domains, including non-integrally closed cases, providing methods to determine solutions explicitly.

Contribution

It generalizes previous results to non-integrally closed domains, offering effective solutions for discriminant equations in broader algebraic settings.

Findings

Finiteness of solutions for discriminant equations over finitely generated domains

Effective methods to compute solution representatives

Extension of results to non-integrally closed 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.