On the multiplicity of zeros of polynomials over arbitrary finite dimensional k-algebras
type: Item
instance of: scholarly article
title: On the multiplicity of zeros of polynomials over arbitrary finite dimensional k-algebras
zbMATH Open document ID: 0581.12011
DOI: 10.1007/BF01215249
author: F. Leon Pritchard
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155045
review text: The author continues the study of zeros begun by \textit{H. Röhrl} [ibid. 25, 359-390 (1978; Zbl 0425.17001)]. Let \(A\subseteq B\) be m-ary algebras over a field k. Let P(x) be in the algebra of formal polynomials over A. Then P(x) has a natural image \(P_ A\) in the algebra of polynomial functions over A. The main results are: (2.2) For k algebraically closed, the number of zeros of \(P_ A\) is either infinite or at most \(n^ d\), where n is the degree of \(P_ A\) and d is the vector space dimension of A. - (4.2) For k infinite, an inequality relating \(m(P_ A:a)\), the multiplicity of a in A as a zero of \(P_ A\), to \(m(P_ B:a)\). - (5.1) Extension of the base field k leaves \(m(P_ A:a)\) unchanged.
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 17A01
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 3929117
zbMATH Keywords: zeros of polynomial
zbMATH Keywords: m-ary algebras
zbMATH Keywords: algebra of formal polynomials
zbMATH Keywords: algebra of polynomial functions
zbMATH Keywords: multiplicity
reviewed by: C. P. L. Rhodes
MaRDI profile type: Publication
cites work: Codimension and multiplicity
cites work: On System of Parameters, Local Intersection Multiplicity and Bezout's Theorem
cites work: Q5542855
cites work: On the zeros of polynomials over arbitrary finite dimensional algebras
cites work: Q3249822
OpenAlex ID: W2048260351
description: scientific article; zbMATH DE number 3929117