Integral Representations and Asymptotics for a Family of Areal Mahler Measures

2026/08/03 by Quanyu Tang, Shu Zhang
Mathematics · #math.NT #msc:11R06 #msc:41A60 #msc:42A38 #msc:60E10

paper · pdf

14 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

We answer a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family P_n(x₁,…,x_n,u) = ∏_j=1ⁿ(1+x_j) + u∏_j=1ⁿ(1-x_j), n≥1. Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for \mathrm m_{\mathbb D}(P_n). These representations yield strict alternating signs for all forward differences of the sequence (\mathrm m_{\mathbb D}(P_n))_n≥1, as well as an explicit three-term asymptotic expansion as n→∞.

Citations

• Areal Weil Heights
• Random walks through the areal Mahler measure: steps in the complex plane
• The Mahler measure of a family of polynomials with arbitrarily many variables
• Evaluations of the areal Mahler measure of multivariable polynomials
• On the Mahler measure of (1+x)(1+y)+z