On the number of polynomials and integral elements of given discriminant
type: Item
instance of: scholarly article
title: On the number of polynomials and integral elements of given discriminant
zbMATH Open document ID: 0653.13005
DOI: 10.1007/BF01903342
published in: Acta Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 11D99
zbMATH DE Number: 4065132
zbMATH Keywords: integral extension ring
zbMATH Keywords: polynomials with given discriminant
author: Jan-Hendrik Evertse
author: Kálmán Győry
reviewed by: Hideo Yokoi
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01903342
OpenAlex ID: W2118715122
description: scientific article; zbMATH DE number 4065132