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Journal Articles Compositio Mathematica Year : 2021

Volume function and Mahler measure of exact polynomials

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.
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Dates and versions

hal-03377099 , version 1 (15-10-2021)

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Antonin Guilloux, Julien Marché. Volume function and Mahler measure of exact polynomials. Compositio Mathematica, 2021, 157 (4), pp.809-834. ⟨10.1112/S0010437X21007016⟩. ⟨hal-03377099⟩
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