大多数$(0,1)$-多面体不是正规的
Most $(0,1)$-polytopes are not normal
- University of California, Davis（加州大学戴维斯分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明随维度增大，$d$维$(0,1)$-多面体的正规类比例以双指数速率趋于零，分类低维多面体并找到首个具旗帜幺模三角剖分但环面环非Koszul的多面体，还给出非正规的δ-拟阵多面体示例。
AI中文摘要：
我们证明，当维度$d$趋向无穷大时，$d$维$(0,1)$-多面体的$0/1$等价类中，正规类的比例至少以双指数速率趋于零。由此可得，对于具有以下三角剖分类型的任意类，该结论同样成立：(a)二次型、(b)旗帜幺模型、(c)正则幺模型或(d)幺模型等。我们针对$d\leq5$的$d$维$(0,1)$-多面体的$0/1$等价类，按其是否允许幺模、旗帜幺模或二次型三角剖分进行分类。在5维情形下，1226525个类中恰好有175个类具有旗帜幺模三角剖分，但无二次型三角剖分；其中存在一些多面体，其环面环并非Koszul环，因此我们找到了首个具有旗帜幺模三角剖分但环面环非Koszul的多面体。与拟阵情形不同，我们展示了一个非正规的δ-拟阵多面体。
英文摘要：
We prove that the proportion of $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes that are normal tends to zero at least at a double exponential rate as $d\to\infty$. As a consequence, the same holds for any of the following classes given by the type of triangulation possible: (a) quadratic, (b) flag unimodular, (c) regular unimodular, or (d) unimodular, among others. We classify the $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes for $d\leq5$ according to whether they admit a unimodular, flag unimodular, or quadratic triangulation. In dimension five, exactly $175$ out of $1{,}226{,}525$ classes have a flag unimodular triangulation, but no quadratic triangulation. Among them, there are polytopes whose toric rings are not Koszul; thus, we find the first polytopes that have a flag unimodular triangulation, but whose toric ring is not Koszul. In contrast with the matroid case, we exhibit a delta-matroid polytope that is not normal.
