# Volume function and Mahler measure of exact polynomials

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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https://hal.archives-ouvertes.fr/hal-01758986
Contributor : Antonin Guilloux <>
Submitted on : Thursday, April 5, 2018 - 7:45:30 AM
Last modification on : Friday, April 10, 2020 - 5:08:56 PM

### Identifiers

• HAL Id : hal-01758986, version 1
• ARXIV : 1804.01395

### Citation

Antonin Guilloux, Julien Marché. Volume function and Mahler measure of exact polynomials. 2018. ⟨hal-01758986⟩

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