AI 中文总结
该研究解答了Matilde Lalín提出的多项式族面积型Mahler测度问题，通过概率重述推导积分表示，得到序列差分符号规律与渐近展开。
AI 中文摘要
我们解答了Matilde Lalín提出的关于多元多项式族$P_n(x_1,\boldsymbol{\rm ldots},x_n,u) = \boldsymbol{\rm prod}_{j=1}^n(1+x_j) + u\boldsymbol{\rm prod}_{j=1}^n(1-x_j)$（$n\boldsymbol{\rm geq}1$）的面积型Mahler测度问题。通过概率重述，我们推导了$\boldsymbol{\rm m}_{\boldsymbol{\rm D}}(P_n)$的卷积与一维傅里叶积分表示，这些表示给出序列$(\boldsymbol{\rm m}_{\boldsymbol{\rm D}}(P_n))_{n\boldsymbol{\rm geq}1}$所有前向差分的严格交替符号，以及$n\to\boldsymbol{\rm infty}$时的显式三项渐近展开。
英文摘要
We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$ P_m(x_1,\ldots,x_m,u) = \prod_{j=1}^m(1+x_j) + u\prod_{j=1}^m(1-x_j), \qquad m\geq1. $$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for $\mathrm m_{\mathbb D}(P_m)$. We prove that, for every fixed $m$, the value $π^m\mathrm m_{\mathbb D}(P_m)$ belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case $m=2$ explicitly in terms of $\operatorname{Li}_4(1/2)$, $ζ(3)$, Catalan's constant, $π$, and $\log2$. We also derive an explicit three-term asymptotic expansion for $\mathrm m_{\mathbb D}(P_m)$ as $m\to\infty$.
Comments28 pages; substantially revised and expanded from v1. Added two main theorems on cyclotomic special values and an explicit formula for m=2
