Multiplicities of degenerations of matrices and mixed volumes of Cayley polyhedra

TL;DR

This paper introduces a method to compute the multiplicity of isolated zeros of maximal minors of matrices composed of generic homogeneous polynomials by leveraging a Pick-type formula for mixed volumes of Cayley polyhedra.

Contribution

It presents a novel application of mixed volume calculations via Cayley polyhedra to determine multiplicities of polynomial matrix zeros.

Findings

Derived a Pick-type formula for mixed volumes of Cayley polyhedra.

Computed multiplicities of zeros for matrices of generic homogeneous polynomials.

Established a link between mixed volumes and algebraic multiplicities.

Abstract

Using a certain Pick-type formula for the mixed volume of Cayley polyhedra, we compute the multiplicity of the isolated common zero of the maximal minors for a matrix of generic homogeneous polynomials of given degrees.