On the irreducibility of multivariate subresultants

Abstract

Let $P_1,...,P_n$ be generic homogeneous polynomials in $n$ variables of degrees $d_1,...,d_n$ respectively. We prove that if $ν$ is an integer satisfying ${\sum_{i=1}^n d_i}-n+1-\min\{d_i\}<ν,$ then all multivariate subresultants associated to the family $P_1,...,P_n$ in degree $ν$ are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of $\binom{ρ-ν+n-1}{n-1}$ smooth isolated points in $\PP^{n-1}.$