Volume function and Mahler measure of exact polynomials

Antonin Guilloux; Julien Marché

doi:10.1112/s0010437x21007016

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arXiv · 1804.01395

Volume function and Mahler measure of exact polynomials

Antonin Guilloux·Julien Marché

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Abstract

We study a class of 2-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 2-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 $K$-theoretical criterium for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.

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BibTeX RIS

Antonin Guilloux, Julien Marché. 2018-04-04. Volume function and Mahler measure of exact polynomials. https://doi.org/10.1112/s0010437x21007016

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