A chain rule for the resultant of two homogeneous polynomials
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.
scientific article; zbMATH DE number 4214162