Gcd of multivariate polynomials via Newton polytopes
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)]).
scientific article; zbMATH DE number 5912645