发表机构
Universidade Federal da Paraíba; Oklahoma State University（帕拉伊巴联邦大学; 俄克拉荷马州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从概率视角研究多项式Bohnenblust--Hille比值，揭示典型系数方向上的尺度与极值常数的鲜明对比，并给出复空间上的一致渐近尺度及实空间上的零测度极限。
AI 中文摘要
多项式 Bohnenblust--Hille 不等式通过其上确界范数控制一个 $m$ 次齐次多项式的系数 $\ell_{2m/(m+1)}$-范数，且常数与维数无关。我们从概率的角度研究相关的 Bohnenblust--Hille 比值，在欧几里得系数球面上放置归一化表面测度。我们的结果揭示了控制经典 Bohnenblust--Hille 常数的极值行为与系数方向上所见典型尺度之间的鲜明对比。对于任意给定的单项式支撑，该比值最终几乎必然至多为 $1$，并且当单项式个数趋于无穷时，它在球面测度下趋于零。在完全复多项式空间上，我们确定了其典型渐近尺度，且该尺度对维数一致；在临界情形 $n_m/m\to1$ 下，这给出 $2\sqrt{2}/\sqrt{m\log m}$，与非消失的极值尺度形成对比。对于实多项式，相应的比值在变量个数趋于无穷时于球面测度下趋于零。
英文摘要
The polynomial Bohnenblust--Hille inequality controls the coefficient $\ell_{2m/(m+1)}$-norm of an $m$-homogeneous polynomial by its supremum norm, with a constant independent of the dimension. We study the associated Bohnenblust--Hille ratio from a probabilistic point of view, by placing normalized surface measure on the Euclidean coefficient sphere. Our results reveal a sharp contrast between the extremal behavior governing the classical Bohnenblust--Hille constants and the typical scale seen in coefficient directions. For arbitrary prescribed monomial supports, the ratio is eventually at most $1$ almost surely, and it tends to zero in spherical measure exactly when the number of monomials tends to infinity. On the full complex polynomial spaces we determine its typical asymptotic scale uniformly in the dimension; in the critical regime $n_m/m\to1$ this gives $2\sqrt2/\sqrt{m\log m}$, in contrast with the nonvanishing extremal scale. For real polynomials, the corresponding ratio tends to zero in spherical measure exactly when the number of variables tends to infinity.
