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arXiv · 2503.22536

Random walks through the areal Mahler measure: steps in the complex plane

Matilde N. Lalín·Siva Sankar Nair·Berend Ringeling·Subham Roy
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Abstract

We study the areal Mahler measure of the two-variable, $k$-parameter family $x+y+k$ and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family $(x+1)(y+1)+kz$ in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family.

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Matilde N. Lalín, Siva Sankar Nair, Berend Ringeling, Subham Roy. 2025-03-28. Random walks through the areal Mahler measure: steps in the complex plane. https://arxiv.org/abs/2503.22536

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