Integral elements with given discriminant over function fields (Q1112876)

Let K:L:k(z) be finite extensions of function fields in one variable with an algebraically closed constant field k of characteristic 0. Let further \({\mathfrak O}_ K\), \({\mathfrak O}_ L\) be the rings of k[z]-integers. The author considers the problem of effectively determining the elements \(\alpha\in {\mathfrak O}_ K\) with a fixed discriminant \(\delta\in {\mathfrak O}_ L\), and generalizations of that problem, e.g. for S-integers. His main result is theorem 1, which gives finiteness statements and effective bounds for the height of a solution \(\alpha\) of the discriminant equation. The introduction, as well as section 3 that presents the results, contains some useful discussion of related problems (e.g., extensions of number fields, or finitely generated integral domains over \({\mathbb{Z}})\), and of the results due to other authors.