二项式边理想与条件独立性
Binomial edge ideals and conditional independence statements
摘要：
引入了单图$G$上的二项式边理想，并研究了它们的代数性质。我们描述了那些图的二次生成元形成一个Gröbner基础的字典顺序诱导的顶点标签。这样的图是弦的和无爪的。给出了一般$G$的一个约化无平方Gröbner基.因此，所有的二项式边理想都是根理想。它们的极小素数可以用$G$的顶点的特定子集来刻画。给出了闭图和非闭图的Cohen-Macaulay性的充分条件。二项式边理想在条件独立理想的研究中自然出现。我们的结果适用于类的条件独立理想，其中一个固定的二进制变量是独立的其他变量的集合，其余的。在这种情况下，准素分解具有自然的统计解释
英文摘要：
We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general $G$. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of $G$. We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation
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