On the number of polynomials and integral elements of given discriminant
Let R be a subring of a field K of characteristic 0 and R' be an integral extension ring of R in a finite normal extension field of K. Denote by \(\Phi\) (R,R') the set of all monic polynomials in R[X] all of whose zeroes are simple and belong to R'. Then, R-equivalence (resp. weakly R- equivalence) relations are defined in R[X], and one of the authors proved in the absolute case (i.e. R is finitely generated over \({\mathbb{Z}})\) that if R is integrally closed in K, the polynomials with given discriminant belong to at most finitely many R-equivalent classes and these classes can be determined effectively [cf. \textit{K. Györy}, J. Reine Angew. Math. 346, 54-100 (1984; Zbl 0519.13008)]. In this paper, the authors intend to establish such effective finiteness theorems in the more general case when \(R^+\) is a subgroup of finite index in \((R'\cap K)^+\), and by use of recent quantitative finiteness results on unit equations they derive explicit upper bounds for the number of R-equivalence classes of polynomials in \(\Phi\) (R,R') with given discriminant both in the absolute and in the relative case (i.e. R is finitely generated over a field algebraically closed in K) [cf. the authors, J. Reine Angew. Math. 358, 6-19 (1985; Zbl 0552.10010)].
scientific article; zbMATH DE number 4065132
scientific article