发表机构
Indian Institute of Technology Bombay（印度理工学院孟买分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究半定松弛分解中因子秩对分解大小的约束，给出置换多面体和正多边形的渐近最小分解阶，并证明秩至多四的多胞形顶点数上界为十二。
AI 中文摘要
我们研究了正半定松弛分解中因子的秩如何约束其大小。我们限制了任何固定分解中低秩顶点因子的数量，并利用这一点确定了在顶点因子秩的任何固定界限下最小分解大小的渐近阶：对于置换多面体 $\Pi_n$ 为 $\Theta(n\log n)$，对于正 $N$ 边形为 $\Theta(\log N)$。在秩为一的情形下，这给出了顶点上实函数空间的最小维数的阶，在该空间中每个面松弛都是平方和，且没有对称性或次数限制。对于无限制的提升，该界给出 $\mathrm{xc}_{\mathrm{PSD}}(\Pi_n)\ge n+\log_3 n-O(1)$，并表明线性大小的置换多面体分解将需要在除消失比例外的所有顶点上具有秩为 $\Omega(\log n)$ 的顶点因子。我们还证明了每个实正半定秩至多为四的多胞形至多有十二个顶点，并为每个平方对称八边形构造了显式的尺寸四提升，因此八是可以达到的。上界结合了对秩为一因子曲线所到达的多边形角点的计数与因子秩上的关联约束。八是否为最大值仍然开放。
英文摘要
We study how the ranks of factors in a positive semidefinite slack factorization constrain its size. We bound the number of low-rank vertex factors in any fixed factorization, and use this to determine the asymptotic order of the minimum factorization size under any fixed bound on vertex-factor ranks: $Θ(n\log n)$ for permutahedra $Π_n$ and $Θ(\log N)$ for regular $N$-gons. In the rank-one case this gives the order of the minimum dimension of a space of real functions on the vertices in which every facet slack is a sum of squares, with no symmetry or degree restriction. For unrestricted lifts, the bound gives $\mathrm{xc}_{\mathrm{PSD}}(Π_n)\ge n+\log_3 n-O(1)$, and shows that a permutahedron factorization of linear size would need vertex factors of rank $Ω(\log n)$ at all but a vanishing fraction of vertices. We also prove that every polytope of real positive semidefinite rank at most four has at most twelve vertices, and construct explicit size-four lifts for every square-symmetric octagon, so eight is attained. The upper bound combines a count of the polygon corners reached by curves of rank-one factors with incidence constraints on factor ranks. Whether eight is the maximum remains open.
