A class of multivariate polynomial convolutions (and applications)

TL;DR

This paper establishes fundamental convolution theorems for multivariate determinantal polynomials, utilizing properties of minor-orthogonal ensembles and mixed discriminants, with applications in permanent computation and singular value addition.

Contribution

It introduces new convolution theorems for multivariate determinantal polynomials and applies them to problems like permanent calculation and unitarily invariant singular value addition.

Findings

Proved two master convolution theorems for multivariate determinantal polynomials.

Rederived a result of Barvinok on permanent computation for low-rank matrices.

Developed a polynomial convolution for unitarily invariant addition of generalized singular values.

Abstract

We prove two "master" convolution theorems for multivariate determinantal polynomials. The methods used include basic properties of what we call a "minor-orthogonal" ensemble as well as properties of the mixed discriminant of matrices. We also give applications, including a rederivation of a result of Barvinok on computing the permanent of a low rank matrix and a polynomial convolution corresponding to the unitarily invariant addition of generalized singular values.