论文标题
Hilbert-Poincaré系列和Gorenstein属性，用于某些非简单多洋粉
Hilbert-Poincaré series and Gorenstein property for some non-simple polyominoes
论文作者
论文摘要
令$ \ mathcal {p} $为没有锯齿形步行的封闭路径，一种非简单的薄薄多粒粉。在本文中，我们对$ k [\ Mathcal {p}] $的$ h $ -polyNomial提供了组合解释，这表明它是$ \ Mathcal {p} $的Rook多项式。 Rinaldo和Romeo（2021）知道，如果$ \ Mathcal {p} $是一个简单的薄型多莫诺，那么$ h $ - 多种元素等于$ \ Mathcal {p} $的Rook多项式，并且认为这属性表征了所有薄型多米诺群。我们的主要示范策略是计算连接到封闭的路径$ \ MATHCAL {p} $的坐标环的简化坐标环，没有Zig-Zag Walks，作为Hilbert-Poincaré系列一系列方便的简单简单的薄有多球体的组合。结果，我们证明了Krull尺寸等于$ \ vert V（\ Mathcal {p}）\ vert - \ Mathrm {rank} \，\ Mathcal {p} $，并且$ k [\ MATHCAL {p}] $的规律性是$ \ Mathcal $ \ Mathcal c} $的ROOK数字。最后，我们表征了Gorenstein Prime闭合路径，证明$ k [\ Mathcal {p}] $是Gorenstein，并且仅当$ \ Mathcal {p} $仅当最大块由三个长度的最大块组成。
Let $\mathcal{P}$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper we give a combinatorial interpretation of the $h$-polynomial of $K[\mathcal{P}]$, showing that it is the rook polynomial of $\mathcal{P}$. It is known by Rinaldo and Romeo (2021), that if $\mathcal{P}$ is a simple thin polyomino then the $h$-polynomial is equal to the rook polynomial of $\mathcal{P}$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert-Poincaré series of the coordinate ring attached to a closed path $\mathcal{P}$ having no zig-zag walks, as a combination of the Hilbert-Poincaré series of convenient simple thin polyominoes. As a consequence we prove that the Krull dimension is equal to $\vert V(\mathcal{P})\vert -\mathrm{rank}\, \mathcal{P}$ and the regularity of $K[\mathcal{P}]$ is the rook number of $\mathcal{P}$. Finally we characterize the Gorenstein prime closed paths, proving that $K[\mathcal{P}]$ is Gorenstein if and only if $\mathcal{P}$ consists of maximal blocks of length three.
