  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.
