The Mahler measure of exact polynomials in three variables

Trieu, Thu Ha

doi:10.2140/ant.2026.20.525

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The Mahler measure of exact polynomials in three variables

Trieu Thu Ha

Vol. 20 (2026), No. 3, 525–576

DOI: 10.2140/ant.2026.20.525

Abstract

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$ -values and Bloch–Wigner dilogarithmic values, conditionally on Beilinson’s conjecture. In some cases, these dilogarithmic values simplify to Dirichlet $L$ -values. The proof involves a construction of an element in $K_{4}^{(3)}$ of a smooth projective curve over a number field. This generalizes a result of Lalín (2015) for the polynomial $z + (x + 1) (y + 1)$ . We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.

Keywords

Mahler's measure, regulator, polylogarithmic complex, motivic cohomology, residue

Mathematical Subject Classification

Primary: 19F27

Secondary: 11G55, 11R06, 19E15

Milestones

Received: 8 April 2024

Revised: 26 December 2024

Accepted: 11 February 2025

Published: 24 March 2026

Authors

Trieu Thu Ha
Unité de Mathématiques Pures et Appliquées École normale supérieure de Lyon 69364 Lyon France

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Vol. 20, No. 3, 2026