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Volume function and Mahler measure of exact polynomials

Part of: Polynomials and matrices

Published online by Cambridge University Press:  14 April 2021

Antonin Guilloux  and
Julien Marché
Antonin Guilloux
Affiliation:
Sorbonne Université, CNRS, IMJ-PRG and INRIA OURAGAN, 75252Paris cédex 05, Franceantonin.guilloux@imj-prg.fr
Julien Marché
Affiliation:
Sorbonne Université, CNRS, IMJ-PRG, 75252Paris cédex 05, Francejulien.marche@imj-prg.fr
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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.

Keywords

Mahler measureA-polynomialamoebaK-theoryhyperbolic geometry

MSC classification

Primary: 11C08: Polynomials
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 4 , April 2021 , pp. 809 - 834
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Copyright
© The Author(s) 2021

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