Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the weaker bound \[ \mathbf{BSA}[f]\le \exp(C_d\sqrt{\log n}). \]

翻译：多项式阈值函数（PTFs）是一类重要的低复杂度布尔函数，与学习理论和逼近理论有密切联系。近期关于PTF学习与测试的研究利用了该类函数的结构性质和等周性质，特别是平均敏感度的界——这是自Gotsman-Linial猜想以来PTF研究的核心主题之一。本文从布尔表面积（或Talagrand边界）\[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)} \]（离散立方体上顶点边界复杂度的自然度量）的角度研究PTF。我们的主要结果是：每个d次PTF具有多对数布尔表面积：\[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d} \]。证明基于Kabanets、Kane和Lu的PTF限制引理 \cite{KKL2017}，并通过点态敏感度的尾界实现。特别地，它控制了敏感度的所有次临界分数阶矩。我们还记录了布尔表面积的随机块划分原理，以及基于Kane关于平均敏感度的工作 \cite{DK} 的另一种递归论证方法，后者独立地得到较弱界 \[ \mathbf{BSA}[f]\le \exp(C_d\sqrt{\log n}) \]。