Gcd of multivariate polynomials via Newton polytopes
type: Item
instance of: scholarly article
title: Gcd of multivariate polynomials via Newton polytopes
zbMATH Open document ID: 1264.12010
author: Luiz Emilio Allem
author: Vilmar Trevisan
published in: Applied Mathematics and Computation
publication date: 2011-06-24T00:00:00Z
review text: In this paper the authors used some geometric properties of Newton polytopes to obtain a criterion establishing that two multivariate polynomials are relatively prime. More precisely, they studied the coprimality between multivariate polynomials verifying whether their associated Newton polytopes have no common factors by showing that they have no common parallel edges. In the latest section a detailed example and a polynomial-time algorithm to compute the gcd between two bivariate polynomials using Newton polygons are presented. One of the first results connecting geometric and algebraic properties of polynomials was done by Ostrowski (1921); and recently, \textit{S. Gao} [J. Algebra 237, 501--520 (2001; Zbl 0997.12001)] used the geometric properties of polytopes to construct families of absolutely irreducible polynomials. Some geometric properties of polytopes are also used to study the indecomposable polynomials (for example, \textit{G. Chéze} and \textit{S. Najib} [J. Algebra 324, No. 1, 1--11 (2010; Zbl 1202.13014)]) and the problem of irreducibility of polynomials after deformation (for example, \textit{A. Bodin, P. Dèbes} and \textit{S. Najib} [Acta Arith. 139, No. 1, 79--100 (2009; Zbl 1228.12002)]).
reviewed by: Salah Najib
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 13P05
zbMATH DE Number: 5912645
zbMATH Keywords: Newton polytope
zbMATH Keywords: multivariate polynomial
zbMATH Keywords: greatest common divisor
describes a project that uses: polymake
MaRDI profile type: Publication
full work available at URL: j.amc.2011.03.035
OpenAlex ID: W2057967145
cites work: Factoring polynomials via polytopes
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cites work: Decomposition of polytopes and polynomials
cites work: Modern computer algebra
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cites work: Q4057549
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DOI: 10.1016/J.AMC.2011.03.035
description: scientific article; zbMATH DE number 5912645