The logarithmic derivative for resultants of systems of algebraic equations (Q1814377)

We consider the system of equations \[ f_ 1(\zeta,w)=0,\;\ldots,\;f_{n+1}(\zeta,w)=0 \tag{1} \] where \(\zeta=(\zeta_ 0,\zeta_ 1,\dots,\zeta_ n)\;\mathbb{C}^{n+1}\), \(w\in\mathbb{C}^ 1\), the functions \(f_ j(\zeta,w)\) are polynomials, which are homogeneous with respect to \(\zeta\). The main restriction upon the system (1) consists in the following: the system (1) has a finite number of solutions \((\zeta^ k,w_ k)\), \(k=1,\dots,M\). We call the polynomial \(p(w)=\prod^ M_{h=1}(w-w_ k)\) the resultant of the system (1) (with respect to \(w)\). The goal of the article is to obtain a formula connecting the logarithmic derivative of the resultant with the sum of global residues of subsystems of the system (1).