Lehmer's conjecture for Hermitian matrices over the Eisenstein and Gaussian integers

TL;DR

This paper proves Lehmer's conjecture for a specific class of polynomials derived from Hermitian matrices over Eisenstein and Gaussian integers, establishing a lower Mahler measure bound of Lehmer's number.

Contribution

It demonstrates that all polynomials from these Hermitian matrices have Mahler measure at least Lehmer's number, confirming Lehmer's conjecture for this class.

Findings

Mahler measure of these polynomials is at least Lehmer's number

Lehmer's conjecture holds for Hermitian matrices over Eisenstein and Gaussian integers

All such polynomials meet the Mahler measure lower bound

Abstract

We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number \tau_0 = 1.17628... .