Interpolation by integer-valued polynomials
type: Item
instance of: scholarly article
title: Interpolation by integer-valued polynomials
zbMATH Open document ID: 0927.13023
author: Sophie Frisch
published in: Journal of Algebra
publication date: 1999-11-18T00:00:00Z
review text: The author pursues two directions to construct interpolating integer-valued polynomials on Krull domains \(R\), that means, given distinct \(a_1, \dots, a_n\in S\leq R\) and \(b_1, \dots, b_n\in R\) there exists an \(f\in \text{Int}(S,R)= \{f\in K[x] \mid f(S)\subseteq R\}\), \(K\) being the quotient field of \(R\), with \(f(a_i) =b_i\), \(i=1, \dots,n\). One approach, running along classical lines, culminates in the following result: An interpolating \(f\in \text{Int} (R,R)\) exists if and only if the \(a_i\) are pairwise incongruent mod all \(P\in\text{Spec}^1(R)\) with \([R:P]= \infty\). The second one is based on so-called weak \(v\)-sequences for \(R\) \((v\) being a valuation of \(R)\) and binomial polynomials constructible from them. Here the corresponding result claims that given an infinite subring \(S\) of \(R\) an \(f\in\text{Int}(S,R)\) exists which interpolates on \(a_1, \dots, a_n\) if this set is a weak \(v\)-sequence for all essential valuations of \(R\). Investigations on the degree of the interpolating polynomial are included.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 1271338
zbMATH Keywords: \(v\)-sequence
zbMATH Keywords: interpolating integer-valued polynomials
zbMATH Keywords: Krull domains
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: jabr.1998.7741
OpenAlex ID: W2059701970
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DOI: 10.1006/JABR.1998.7741
description: scientific article; zbMATH DE number 1271338