两类 Hermite 标准形单纯形

中文摘要 AI 辅助

我们研究了 Bruckamp、Caicedo 和 Juhnke 所研究的两类 Hermite 标准形单纯形的 Ehrhart 系数和整数分解性质。第一类由形如 $S_{\boldsymbol{a}}=\mathrm{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol{a})$ 的单纯形组成，其中 $\boldsymbol{a}=(a_1,\ldots,a_{d-1},N)$；第二类包含单纯形 $T_{d,N}=\mathrm{conv}\bigl(0,e_1,\ldots,e_{d-2},(d-2,\ldots,d-2,d-1,0),(1,\ldots,1,N)\bigr)$。对于族 $T_{d,N}$，我们在任意维度下建立了 Ehrhart 系数的单峰性，并完全分类了对数凹和实根情形。对于子族 $S_{\boldsymbol{a}}$ 且 $\boldsymbol{a}=(N-q,\ldots,N-q,N)$，我们通过负连分数的同余条件刻画了整数分解性质和正则单模三角剖分的存在性。更一般地，我们将整数分解性质和单模三角剖分的分析推广到任意向量 $\boldsymbol{a}$ 的 $S_{\boldsymbol{a}}$。作为结果，我们的结论解决了 Bruckamp、Caicedo 和 Juhnke 提出的三个开放问题。

英文摘要

We study Ehrhart coefficients, the integer decomposition property, and unimodular triangulations for two families of Hermite normal form simplices considered by Bruckamp, Caicedo, and Juhnke. The first family consists of simplices of the form $S_{\boldsymbol{a}}=\mathrm{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol{a})$, where $\boldsymbol{a}=(a_1,\ldots,a_{d-1},N)$, and the second consists of the simplices $T_{d,N}=\mathrm{conv}\bigl(0,e_1,\ldots,e_{d-2},(d-2,\ldots,d-2,d-1,0),(1,\ldots,1,N)\bigr)$. For the family $T_{d,N}$, we prove unimodality of the Ehrhart coefficients in every dimension and determine the log-concave and real-rooted cases. For the subfamily $S_{\boldsymbol{a}}$ with $\boldsymbol{a}=(N-q,\ldots,N-q,N)$, we characterize both the integer decomposition property and the existence of a regular unimodular triangulation via a congruence condition on a negative continued fraction. We also give an arithmetic test for the integer decomposition property and finite triangulation tests for $S_{\boldsymbol{a}}$ with arbitrary $\boldsymbol{a}$. These results address three open questions of Bruckamp, Caicedo, and Juhnke.

发表机构

• Center for Combinatorics, LPMC,Nankai University（南开大学组合数学中心）
• School of Mathematical Sciences,Capital Normal University（首都师范大学数学科学学院）
• School of Mathematics and Statistics,Beijing Institute of Technology（北京理工大学数学与统计学院）

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