Resultant of an equivariant polynomial system with respect to the symmetric group

TL;DR

This paper proves that the resultant of a symmetric polynomial system can be decomposed into smaller resultants using divided differences, and applies this to derive a formula for the discriminant of symmetric polynomials.

Contribution

It introduces a decomposition method for the resultant of symmetric polynomial systems based on divided differences, providing new insights into their structure.

Findings

Resultant decomposes into smaller resultants via divided differences

Derived a formula for the discriminant of symmetric polynomials

Enhanced understanding of symmetric polynomial systems

Abstract

Given a system of n homogeneous polynomials in n variables which is equivariant with respect to the canonical actions of the symmetric group of n symbols on the variables and on the polynomials, it is proved that its resultant can be decomposed into a product of several smaller resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial.