An extension of elementary symmetric polynomials and power sums to the case of multivariate polynomials (Q2765096)

The problem of relating the roots of a multivariate polynomial with its coefficients is investigated in this paper. A technique consists in considering a special type of multivariate interpolation polynomials introduced by \textit{P. Kergin} [J. Approximation Theory 29, 278-293 (1980; Zbl 0492.41008)]. Natural extensions of multivariate elementary symmetric polynomials and power sums are given to derive a formula which is similar to the Newtonian equations that relate the univariate elementary symmetric polynomials to the power sums. The author proves that one of the fundamental properties which are shared by both the univariate and multivariate cases, is that the multivariate elementary symmetric polynomials \(\sigma_m\), are generated by the function \(E(x)= \prod^\mu_{i=0} (1+x^{(i)}\dot x)\). Another result is that the generating function \(P\) of the univariate power sums can be generalized to multivariate power sums as \(P(x)=\text{grad} E(-x)/E(-x)\).