Volume function and Mahler measure of exact polynomials. (14th April 2021)

• Record Type: Journal Article
• Authors: Guilloux, Antonin; Marché, Julien
• Is Part Of: Compositio mathematica. Volume 157:Number 4(2021)
• Page Start: 809
• Page End: 834
• Publication Date: 2021-04-14
• Subjects: Mahler measure, -- A-polynomial, -- amoeba, -- K-theory, -- hyperbolic geometry; 11C08; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007016
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.