Resultant of an equivariant polynomial system with respect to a direct product of $r$ symmetric groups

Abstract

In this paper we study the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of a direct product of symmetric groups. We first treat, in detail, the case of a product of two symmetric groups, and establish a decomposition formula for the resultant of such systems. We then show that this decomposition, together with the underlying combinatorics, extends to an arbitrary (finite) direct product of $r$ symmetric groups. Thanks to these decomposition formulas, we prove that the discriminant of a multivariate homogeneous polynomial invariant under a direct product of $r$ symmetric groups splits into a product of resultants of smaller size that are easier to compute.