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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2014
- DOI:
- 10.48550/arXiv.1410.0064
- arXiv:
- arXiv:1410.0064
- Bibcode:
- 2014arXiv1410.0064H
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Combinatorics;
- Mathematics - Group Theory
- E-Print:
- 3 figures