Quadratic Gröbner bases of twinned order polytopes

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• 作者：Takayuki Hibi ^{hibi@math.sci.osaka-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Kazunori Matsuda ^{kaz-matsuda@math.sci.osaka-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：54
• 期：Complete
• 页码：187-192
• 全文大小：357 K

文摘

Let P$P$ and Q$Q$ be finite partially ordered sets on [d]={1,…,d}$\left[d\right]=\{1,\dots ,d\}$, and O(P)⊂R^d$\mathcal{O}\left(P\right)\subset {\mathbb{R}}^d$ and O(Q)⊂R^d$\mathcal{O}\left(Q\right)\subset {\mathbb{R}}^d$ their order polytopes. The twinned order polytope of P$P$ and Q$Q$ is the convex polytope Δ(P,−Q)⊂R^d$\Delta (P,-Q)\subset {\mathbb{R}}^d$ which is the convex hull of O(P)∪(−O(Q))$\mathcal{O}\left(P\right)\cup (-\mathcal{O}\left(Q\right))$. It follows that the origin of R^d${\mathbb{R}}^d$ belongs to the interior of Δ(P,−Q)$\Delta (P,-Q)$ if and only if P$P$ and Q$Q$ possess a common linear extension. It will be proved that, when the origin of R^d${\mathbb{R}}^d$ belongs to the interior of Δ(P,−Q)$\Delta (P,-Q)$, the toric ideal of Δ(P,−Q)$\Delta (P,-Q)$ possesses a quadratic Gröbner basis with respect to a reverse lexicographic order for which the variable corresponding to the origin is the smallest. Thus in particular if P$P$ and Q$Q$ possess a common linear extension, then the twinned order polytope Δ(P,−Q)$\Delta (P,-Q)$ is a normal Gorenstein Fano polytope.