论文标题
欧拉理想
Eulerian ideals
论文作者
论文摘要
令$ g $为简单的图表，$ i（x_g）=φ^{ - 1}（x_i^2-x_j^2：i，j \ in v_g）$，其中$φ\ colon k [e_g] \ to k [v_g] $是向其Vertices的产品发送边缘的同构。理想的$ i（x_g）$是cohen--macaulay，一维和二项式。如果$ g $是双方的，那么众所周知，$ i（x_g）$的穆斯福尔诺沃（Mumford）规律性等于一组边缘的最大基数，其中$ g $的任何Eulerian子图的边缘的一半不超过一半。在这里，关于与$ g $的边缘订单相关的GREVLEX订单，我们描述了$ i（x_g）$的gröbner基础，我们在偶数标有帕尔特式标记的顶点方面表征了理想$（i（x_g），t_e）$的标准单元。使用这些结果，我们通过$ g $的一组均匀的顶点对$ i（x_g）$的度进行了组合解释；而且我们表明，对于任何图，Castelnuovo- Mumford $ i（x_g）$的规律性是一组边缘的最大基础性，这些边缘的边缘不超过$ g $的任何\ emph {偶} $ g $的边缘的一半，或者是相当于，最大的固定固定代码$ t $ t $ t $ t $ -jjoin的最大含量。
Let $G$ be a simple graph and $I(X_G)=φ^{-1}(x_i^2-x_j^2 : i,j\in V_G)$, where $φ\colon K[E_G]\to K[V_G]$ is the homomorphism that sends an edge to the product of its vertices. The ideal $I(X_G)$ is Cohen--Macaulay, one-dimensional and binomial. If $G$ is bipartite, it is known that the Castelnuovo--Mumford regularity of $I(X_G)$ is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of $G$. Here, with respect to the grevlex order associated to an ordering of the edge set of $G$, we describe a Gröbner basis for $I(X_G)$, and we characterize the standard monomials of the ideal $(I(X_G),t_e)$ in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of $I(X_G)$, via the set of even sets of vertices of $G$; and we show that the Castelnuovo--Mumford regularity of $I(X_G)$, for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any \emph{even} Eulerian subgraph of $G$ or, equivalently, the maximum cardinality of a minimum fixed parity $T$-join.
