# Resultant of an equivariant polynomial system with respect to direct product of symmetric groups

## TL;DR

This paper introduces a decomposition formula for the resultant of equivariant polynomial systems under direct product symmetric groups, simplifying the computation of discriminants of invariant polynomials.

## Contribution

It provides a novel decomposition formula for resultants of equivariant polynomial systems under direct product symmetric groups, enabling easier discriminant calculations.

## Key findings

- Decomposition formula for resultants established
- Discriminants split into smaller, computable resultants
- Simplifies analysis of invariant multivariate polynomials

## Abstract

In this note, we consider the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of direct product of two symmetric groups. We establish a decomposition formula for the resultant of such systems. Thanks to that decomposition formula we prove that the discriminant of an invariant multivariate homogeneous polynomial under a direct product of symmetric groups splits into smaller resultants that are easier to compute.