发表机构
RIKEN iTHEMS（理化学研究所 iTHEMS）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明简单有向拟阵的顶部图幅值等于其底层拟阵动机zeta函数在$x=-1$的特殊化，细化Las Vergnas--Zaslavsky定理，构造反例反驳Koizumi--Liu猜想，并建立Varchenko--Gelfand滤过以恢复完整动机zeta函数。
AI 中文摘要
我们证明了简单有向拟阵的顶部图的幅值是其底层拟阵的动机zeta函数在$x=-1$处的特殊化。这细化了Las Vergnas--Zaslavsky定理，并为任意拟阵定义了幅值。对于简单可定向拟阵，我们根据平坦链计算了在$q=-1$处极点的阶数，并将其与顶部奇偶函数的Varchenko--Gelfand度等同。通过比较极点阶数，我们构造了一个秩为六的实排列$\nmathcal A$，使得$\operatorname{Mag}(\mathcal A;-t)$具有无穷多个负系数，从而反驳了Koizumi--Liu的最终符号交替猜想。随后，我们在简单有向拟阵的模2幅值上同调上构造了一个典范的乘法Varchenko--Gelfand滤过，并证明其分次维数恢复了完整的动机zeta函数。
英文摘要
We prove that the magnitude of the tope graph of a simple oriented matroid is the specialization at $x=-1$ of the motivic zeta function of its underlying matroid. This refines the Las Vergnas--Zaslavsky theorem and defines magnitude for arbitrary matroids. For simple orientable matroids, we compute the order of the pole at $q=-1$ from chains of flats and identify it with the Varchenko--Gelfand degree of the tope parity function. Comparing pole orders, we construct a rank-six real arrangement $\mathcal A$ such that $\operatorname{Mag}(\mathcal A;-t)$ has infinitely many negative coefficients, disproving Koizumi--Liu's eventual sign alternation conjecture. We then construct a canonical multiplicative Varchenko--Gelfand filtration on the mod-$2$ magnitude cohomology of a simple oriented matroid and show that its graded dimensions recover the full motivic zeta function.
Comments33 pages. Comments welcome!
