论文标题
多米诺骨牌的二项式理想
Binomial ideals of domino tilings
论文作者
论文摘要
在本文中，我们考虑了共有群落区域的所有多米诺骨牌的集合。我们探索的主要问题是：我们如何从一个瓷砖转移到另一个瓷砖？瓷砖空间可以看作是具有固定度序列的固定图的子图的空间。在代数统计中探索了连接此类空间的动作。因此，我们从应用的代数观点来解决这个问题，从而在多米诺骨牌瓷砖，代数统计和曲折代数之间建立了新的联系。利用图形的曲折理想的结果，我们能够描述连接任何维度的给定立方体区域的平铺空间的移动。这是通过研究由同一区域的两个不同多米诺骨牌引起的二项式来完成的。此外，我们介绍了瓷砖理想和翻转理想，并使用这些理想来重述连接瓷砖空间的含义。最后，我们表明，如果$ r $是$ 2 $二维的简单连接的立方体区域，则可以用二元二元组来写的任何二线二元式，以二元性二元组来编写。作为我们主要结果的推论，我们获得了以下事实的替代证明：$ 2 $二维的简单连接区域的多米诺骨牌瓷砖集通过翻转连接。
In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if $R$ is a $2$-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of $R$ can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a $2$-dimensional simply connected region is connected by flips.
