Tropical Laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions

TL;DR

This paper extends tropical root theory from polynomials to Laurent series, establishing a bijection with Newton polygon slopes and applying it to eigenvalue localization of matrix functions.

Contribution

It introduces a new definition of tropical roots for Laurent series, explores their properties, and applies the theory to eigenvalue localization and convergence radii.

Findings

Tropical roots of Laurent series can be infinite in number.

There can be at most two tropical roots with infinite multiplicity.

The theory relates tropical roots to convergence radii and eigenvalue localization.

Abstract

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton polygon associated with the tropical Laurent series. We show that, unlike in the polynomial case, there may be infinitely many tropical roots; moreover, there can be at most two tropical roots of infinite multiplicity. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.