Polynomial convolutions in max-plus algebra

TL;DR

This paper extends polynomial convolutions from classical algebra to max-plus algebra, providing explicit root descriptions and connecting to combinatorial and probabilistic concepts in a novel algebraic setting.

Contribution

It introduces max-plus algebra analogues of polynomial convolutions that preserve canonical forms and offer explicit root characterizations.

Findings

Max-plus polynomial convolutions have explicit root descriptions.

The max-permanent replaces the determinant in this setting.

Results connect combinatorial polynomials with max-plus algebra.

Abstract

Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava and then Marcus studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability. We explore analogues of these types of convolutions in the setting of max-plus algebra. In this setting the max-permanent replaces the determinant, the maximum is the analogue of the expected value and real-rootedness is replaced by full canonical form. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots.