The spectra of polynomial equations with varying exponents

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, Teichmüller 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.

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