Polynomial-mappings and \(M\)-equivalence
type: Item
instance of: scholarly article
title: Polynomial-mappings and \(M\)-equivalence
zbMATH Open document ID: 1106.13007
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: Let \(R\) be an integral domain with quotient field \(K\), and for \(A\subset R\) let \(\text{Int}(A,R)\) be the set of all polynomials \(f\in K[X]\) with \(f(A)\subset R\). \textit{R. Gilmer} and \textit{W. Smith} [Arch. Math. 73, 355--365 (1999; Zbl 0955.13009)] defined two subsets \(S,T\) of \(R\) to be polynomially equivalent if \(\text{Int}(S,R)= \text{Int}(T,R)\), and asked, whether the following holds: if \(R\) is a Dedekind domain, \(E\subset R\) is infinite, \(f\in \text{Int}(E,R)\) and the sets \(E\) and \(f(E)\) are polynomially equivalent, then \(f\) is linear. The author gives a positive answer in the following two cases: a) \(K\) is either an algebraic number field or a finitely generated extension of a finite field, and \(R\) is finitely generated, and b) \(R\) has only finitely many invertible elements, and all its residue fields are finite. Moreover he introduces the degree of exceptionality (\(\text{dex}(R)\)) of a ring \(R\) defined as the least positive integer \(d\) with the property that for every \(R\)-polynomial of degree exceeding \(d\) the following holds: if \(E\subset R\) has the property that for all distinct \(a,b\in E\) the ratio \((f(a)-f(b))/(a-b)\) is invertible, then \(E\) is finite. Various sufficient conditions for \(\text{dex}(R)=1\) are established, and it is shown, in particular, that this holds if \(A\) is a finitely generated domain over a finite field.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 12D99
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 5078153
zbMATH Keywords: polynomial mapping
author: S. B. Mulay
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.04.027
OpenAlex ID: W2002167918
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DOI: 10.1016/J.JALGEBRA.2006.04.027
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description: scientific article; zbMATH DE number 5078153