The spectra of polynomial equations with varying exponents

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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