The spectra of polynomial equations with varying exponents

TL;DR

This paper investigates how solutions to polynomial equations with variable exponents depend on those exponents, with applications to graph theory, 3-manifolds, and free-by-cyclic groups, providing spectral descriptions in these contexts.

Contribution

It introduces a framework for analyzing the spectra of polynomial equations with varying exponents, connecting to diverse mathematical areas and offering new spectral descriptions.

Findings

Spectral descriptions of Alexander polynomials for fibered 3-manifolds

Analysis of Teichmüller polynomials in related contexts

Characterization of characteristic polynomials for directed graphs

Abstract

We study the dependence of solutions of equations of the form $a_0 + a_1 z^{\ell_1} + ... + a_m z^{\ell_m} = 0$, on the exponents $\ell_1, ..., \ell_m$. We apply our results to equations that appear in graph theory, the theory of 3-manifolds fibering over the circle, and the theory of free-by-cyclic groups. In particular, we provide descriptions of the spectra of the Alexander polynomial of a fibered 3-manifold, Teichm\"uller polynomials associated to such a manifold or to a free by cyclic group, and the family of characteristic polynomials of a fixed directed graph with varying edge lengths.