发表机构
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明秩二拟阵在特定条件下Ehrhart $h^*$-多项式的实根性，给出反例否证猜想，并证明超对数凹性及单峰性。
AI 中文摘要
我们证明了当秩二拟阵的最小平行类大小至多为3时，若其恰好具有三个平行类，则其Ehrhart $h^*$-多项式是实根的。这一界限是精确的：对于所有足够大的整数$a$，具有平行类大小$(4,561,600)$和$(4,a,a+29)$的秩二拟阵，其$h^*$-多项式不是实根的。这些反例否证了Ferroni的实根性猜想。它们的对偶是theta图的圈拟阵，并具有相同的$h^*$-多项式。尽管如此，每个秩为二或余秩为二的拟阵都有一个正的$h^*$-系数序列，该序列是阶数等于多项式次数的超对数凹的。特别地，单峰性猜想在这两种情况下都成立。
英文摘要
We prove that the Ehrhart $h^*$-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and $(4,a,a+29)$, for all sufficiently large integers $a$, have $h^*$-polynomials that are not real-rooted. These counterexamples disprove Ferroni's real-rootedness conjecture. Their duals are cycle matroids of theta graphs and have the same $h^*$-polynomials. Nevertheless, every matroid of rank two or corank two has a positive $h^*$-coefficient sequence that is ultra log-concave of order equal to the polynomial's degree. In particular, the unimodality conjecture holds in both cases.
Comments21 pages
