House of algebraic integers symmetric about the unit circle

TL;DR

This paper establishes a new lower bound for the maximum modulus of roots of monic integer polynomials with roots symmetric about the unit circle, extending previous conjectures and utilizing advanced rationality and capacity estimates.

Contribution

It introduces an improved lower bound based on enhanced rationality techniques and weighted Chebyshev constants, confirming Boyd's conjecture.

Findings

New lower bound for roots' maximum modulus

Extension of Dimitrov's work on Schinzel-Zassenhaus conjecture

Confirmation of Boyd's conjecture

Abstract

We give a Schinzel-Zassenhaus-type lower bound for the maximum modulus of roots of a monic integer polynomial with all roots symmetric with respect to the unit circle. Our results extend a recent work of Dimitrov, who proved the general Schinzel-Zassenhaus conjecture by using the P\'olya rationality theorem for a power series with integer coefficients, and some estimates for logarithmic capacity (transfinite diameter) of sets. We use an enhancement of P\'olya's result obtained by Robinson, which involves Laurent-type rational functions with small supremum norms, thereby replacing the logarithmic capacity with a smaller quantity. This smaller quantity is expressed via a weighted Chebyshev constant for the set associated with Dimitrov's function used in Robinson's rationality theorem. Our lower bound for the house confirms a conjecture of Boyd.