Lehmer's conjecture for matrices over the ring of integers of some imaginary quadratic fields

TL;DR

This paper proves Lehmer's conjecture for matrices over certain imaginary quadratic integer rings, establishing a lower bound on Mahler measures for noncyclotomic matrices.

Contribution

It extends Lehmer's conjecture to matrices over rings of integers in specific imaginary quadratic fields, a novel generalization in algebraic number theory.

Findings

Noncyclotomic $R$-matrices have Mahler measure at least 1.176

Lehmer's conjecture holds for these matrices over specified quadratic fields

The result applies to reciprocal polynomials associated with these matrices.

Abstract

Let $R=\mathcal{O}_{\Q(\sqrt{d})}$ for $d<0$, squarefree, $d\neq -1,-3$. We prove Lehmer's conjecture for associated reciprocal polynomials of $R$-matrices; that is, any noncyclotomic $R$-matrix has Mahler measure at least $\lambda_0=1.176...$.