Non-archimedean valuations of eigenvalues of matrix polynomials

TL;DR

This paper explores the relationship between eigenvalues of matrix polynomials over Puiseux series and their tropical analogues, establishing inequalities and conditions for equality, and linking eigenvalues to auxiliary matrix polynomials.

Contribution

It introduces general weak majorization inequalities connecting eigenvalues over Puiseux series with tropical eigenvalues, and characterizes eigenvalues via auxiliary matrix polynomials.

Findings

Weak majorization inequalities relating eigenvalues and tropical analogues

Equality conditions under genericity assumptions

Leading coefficients determined by auxiliary matrix polynomials

Abstract

We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials.