The spectra of polynomial equations with varying exponents

Motivated by results of McMullen about the Teichmüller polynomial, we study the dependence of solutions of equations of the form a0+a1zℓ1+…+amzℓm=0a0+a1zℓ1+…+amzℓm=0, on the exponents ℓ1,…,ℓmℓ1,…,ℓm. We apply our results to equations that appear in the theory of 3-manifolds fibering over the circle, and the theory of free-by-cyclic groups, and in graph theory. In particular, we provide descriptions of the roots of the Alexander polynomial of a fibered 3-manifold, the 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.