Resultant of an equivariant polynomial system with respect to the reflection group $G(r,n)$

Abstract

We consider systems of homogeneous multivariate polynomials equivariant under the complex reflection group $G(r,n)=(\mathbb{Z}/r\mathbb{Z}{)}^n⋊S_n$. Using divided differences indexed by partitions of $n$, we establish a decomposition formula expressing the resultant of such a system as a product of resultants of smaller, partition-indexed subsystems. Combining this with the classical resultant--discriminant relation, we show that the discriminant of a $G(r,n)$-invariant homogeneous polynomial splits explicitly into a product of resultants of smaller subsystems, considerably easier to compute; we illustrate both decompositions with worked examples.