Quanyu Tang

Integral Representations and Asymptotics for a Family of Areal Mahler Measures

We answer a problem posed by Matilde Lal\'in concerning the areal Mahler measures of the multivariable polynomial family $$ P_n(x_1,\ldots,x_n,u) = \prod_{j=1}^n(1+x_j) + u\prod_{j=1}^n(1-x_j), \qquad n\geq1. $$ 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 $\bigl(\mathrm m_{\mathbb D}(P_n)\bigr)_{n\geq1}$, as well as an explicit three-term asymptotic expansion as $n\to\infty$.