Factorization of polynomials in hyperbolic geometry and dynamics

TL;DR

This paper applies polynomial factorization techniques in hyperbolic geometry to compute trace fields and minimal polynomials related to Dehn fillings and pseudo-Anosov maps, advancing understanding of geometric structures.

Contribution

It introduces a method leveraging polynomial factorization to determine trace fields and minimal polynomials in hyperbolic 3-manifolds and mapping classes.

Findings

Computed trace fields for Dehn fillings of the Whitehead link

Determined minimal polynomials of small dilatation pseudo-Anosov maps (assuming Lehmer's Conjecture)

Established conditions on degrees of trace fields over Q

Abstract

Using factorization theorems for sparse polynomials, we compute the trace field of Dehn fillings of the Whitehead link, and (assuming Lehmer's Conjecture) the minimal polynomial of the small dilatation pseudo-Anosov maps and the trace field of fillings of the figure-8 knot. These results depend on the degrees of the trace fields over Q being sufficiently large.