Article contents
Volume function and Mahler measure of exact polynomials
Published online by Cambridge University Press: 14 April 2021
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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- Research Article
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- © The Author(s) 2021
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