发表机构
Antai College of Economics and Management, Shanghai Jiao Tong University（上海交通大学安泰经济与管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过曲线路径编码改进 Kalai--Kleitman 递推，将多面体直径上界的指数降低一个对数因子，并在特定区域获得更优或近线性的界，且证明前导项在该框架内是精确的。
AI 中文摘要
设 $\Delta_u(d,n)$ 为具有 $n$ 个面的尖 $d$ 维多面体的最大图直径。利用迭代 Kalai--Kleitman 递推的曲线路径编码，我们证明，在 $n\geq d\geq 4$ 上一致地有 $\Delta_u(d,n) \leq (n-d)^{\log_2 G_d}$，其中 $G_d = (4\ln 2+o(1))\frac{d}{(\ln d)^2}$，这里 $G_d$ 的渐近表达式理解为 $d\to\infty$ 时的情形，这比之前最佳拟多项式界的指数多改进了一个对数因子。借助定量的 $d$ 步约简，我们还得到互补的基于过剩的界 $\Delta_u(d,n) \leq (n-d)^{\frac{1}{2}\log_2(n-d)+O(1)}$，其中隐含常数为绝对常数。在 $n - d = \Theta(d)$ 的区域，后者渐近更强，并将指数中的首项系数减半。我们进一步考察当 $n$ 相对于 $d$ 增长时这些界的行为，在固定 $0<\gamma<1$ 的 $n = d^{1/\gamma+o(1)}$ 情形获得更尖锐的指数，并在深尾区域 $(\ln n)/d\to\infty$ 获得近线性界。我们还表明，在该正路径计数框架内，一般界的前导项是精确的。
英文摘要
Let $Δ_u(d,n)$ be the maximum graph diameter of a pointed $d$-dimensional polyhedron with $n$ facets. Using a curved-path encoding of the iterated Kalai--Kleitman recurrence, we prove, uniformly over $n\geq d\geq 4$, \[ Δ_u(d,n) \leq (n-d)^{\log_2 G_d},\qquad G_d = (4\ln 2+o(1))\frac{d}{(\ln d)^2}, \] where the asymptotic expression for $G_d$ is understood as $d\to\infty$, improving the exponent of the previous best quasi-polynomial bound by an additional logarithmic factor. With the quantitative $d$-step reduction, we also obtain the complementary excess-based bound \[ Δ_u(d,n) \leq (n-d)^{\frac{1}{2}\log_2(n-d)+O(1)}, \] where the implied constant is absolute. In the regime $n - d = Θ(d)$, the latter bound is asymptotically stronger and halves the leading coefficient in the exponent. We further examine their behavior as $n$ grows relative to $d$, obtaining sharper exponents when $n = d^{1/γ+o(1)}$ for fixed $0<γ<1$ and an almost-linear bound in the deep-tail regime $(\ln n)/d\to\infty$. We also show that the leading term of the general bound is sharp within this positive path-counting framework.
