arXiv 1909.027732019-09-06math.ACmath.AGmath.CO

Joins，Ears和Castelnuovo-Mumford正则性

Joins, Ears and Castelnuovo-Mumford regularity

Jorge Neves, Maria Vaz Pinto, Rafael H. Villarreal

摘要：

我们引入了一类新的多项式理想与一个简单的图，$G$。设$K[E_G]$是$G$的边上的多项式环，$K[V_G]$是$G$的顶点上的多项式环.我们将$G$与理想$I(X_G)$相关联，$I(X_G)$通过映射$K[E_G]\to K[V_G]$定义为$(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$的原像，该映射将与边$e=\{i,j\}$相关联的变量$t_e$发送到与其顶点相关联的变量的乘积$x_ix_j$。证明了$K[E_G]/I(X_G)$是一个一维Cohen-Macaulay分次环，$I(X_G)$是一个二项式理想，并且对于一个固定的单项式阶，它的初始理想有一个独立于场$K$的生成集.我们专注于Castelnuovo-Mumford正则性的$I(X_G)$提供以下尖锐的上限和下限：$$ μ(G) \leq \operatorname{reg} I(X_G) \leq |V_G|-b_0(G)+1, $$其中$μ(G)$是最大顶点连接数的图和$b_0(G)$是其连接组件的数量。我们证明了一个二分图的下界，并利用这个下界得到了一个新的组合结果的偶数长度的巢耳分解的耳朵的数量。

英文摘要：

We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices. We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field $K$. We focus on the Castelnuovo-Mumford regularity of $I(X_G)$ providing the following sharp upper and lower bounds: $$ μ(G) \leq \operatorname{reg} I(X_G) \leq |V_G|-b_0(G)+1, $$ where $μ(G)$ is the maximum vertex join number of the graph and $b_0(G)$ is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

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摘要：

英文摘要：

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