On a conjecture by Boyd

TL;DR

This paper proves a conjectured Mahler measure identity involving two specific Laurent polynomials by establishing relationships between their regulators, advancing understanding in number theory and algebraic geometry.

Contribution

It provides a proof of Boyd's conjecture on Mahler measures by linking regulators of related algebraic curves, a novel approach in this context.

Findings

Confirmed the Mahler measure identity for the given polynomials.

Established relationships between regulators of the associated curves.

Contributed to the theory connecting Mahler measures and algebraic regulators.

Abstract

The aim of this note is to prove the Mahler measure identity $m(x+x^{-1}+y+y^{-1}+5) = 6 m(x+x^{-1}+y+y^{-1}+1)$ which was conjectured by Boyd. The proof is achieved by proving relationships between regulators of both curves.