A chain rule for the resultant of two homogeneous polynomials
type: Item
instance of: scholarly article
title: A chain rule for the resultant of two homogeneous polynomials
zbMATH Open document ID: 0734.12001
DOI: 10.1007/BF01198221
author: James H. McKay
author: Stuart Sui-Sheng Wang
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: Let \(K\) be a field and \(F,G\in K[x,y]\) be homogeneous polynomials in two variables, \[ F(x,y)=f_0x^n+f_1x^{n-1}y+\ldots+f_ny^n, \] \[ G(x,y)=g_0x^m+g_1x^{m-1}y+\ldots+g_my^m. \] Their resultant \(\operatorname{Res}_{x,y}(F,G)\) is the \((m+n)\times (m+n)\)-determinant where the \(i\)-th row has the form \((0,\ldots,0_i,f_0,\ldots,f_n,0,\ldots,0)\) \((0\le i\le m)\) and the \((m+j)\)-th row the form \((0,\ldots,0_j,g_0,\ldots,g_m,0,\ldots,0)\) for \(0\le j\le n\). The main result of the paper is a kind of chain rule for \(\operatorname{Res}\), namely \[ \operatorname{Res}_{u,v}(F(H,K),G(H,K)) = (\operatorname{Res}_{x,y}(F,G))^{\deg(H)}(\operatorname{Res}_{u,v}(H,K))^{\deg (F)\deg (G)}, \] where \(H(u,v)\), \(K(u,v)\) are homogeneous polynomials of same degree. There exist some exceptional cases if only one of both sides equals \(0\). Some consequences of that result, in particular for the resultant of polynomials in one variable are derived, too.
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 4214162
zbMATH Keywords: homogeneous polynomials
zbMATH Keywords: resultant
zbMATH Keywords: chain rule
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
cites work: Resultants of Cyclotomic Polynomials
cites work: Q5827465
cites work: An inversion formula for two polynomials in two variables
cites work: A chain rule for the resultant of two polynomials
description: scientific article; zbMATH DE number 4214162