超图的Alexander对偶的平方的相伴素数
Associated Primes of the Square of the Alexander Dual of Hypergraphs
摘要：
本文的目的是提供确定m-超图H的边理想的Alexander对偶的平方的伴随素数的方法。基于m-超图上的组合条件，证明了边理想的Alexander对偶的平方的相伴素数的一般判定方法。此外，我们证明了一个更有效的组合标准，用于检测非最小相关素数的不存在。在3-超图的研究中，我们证明了以前发现的2-超图（单图）的结果的一个令人惊讶的扩展。对于2-超图，边理想的Alexander对偶的平方的相关素数要么是高度为2，要么是高度大于2的奇数。然而，我们证明，在3-超图的情况下，没有这样的限制-或实际上任何限制-的高度相关联的素数。进一步地，我们把这个结果推广到任何大于大于3的维数。具体地说，给定任意整数m，q和n，其中3 \leq m \leq q \leq n，我们构造一个大小为n的m-超图，其相关素数的高度为q。我们进一步证明了在相同的条件下构造连通m-超图是可能的。
英文摘要：
The purpose of this paper is to provide methods for determining the associated primes of the square of the Alexander dual of the edge ideal for an m-hypergraph H. We prove a general method for detecting associated primes of the square of the Alexander dual of the edge ideal based on combinatorial conditions on the m-hypergraph. Also, we demonstrate a more efficient combinatorial criterion for detecting the non-existence of non-minimal associated primes. In investigating 3-hypergraphs, we prove a surprising extension of the previously discovered results for 2-hypergraphs (simple graphs). For 2-hypergraphs, associated primes of the square of the Alexander dual of the edge ideal are either of height 2 or of odd height greater than 2. However, we prove that in the 3-hypergraph case, there is no such restriction - or indeed any restriction - on the heights of the associated primes. Further, we generalize this result to any dimension greater than 3. Specifically, given any integers m, q, and n with 3\leq m\leq q\leq n, we construct a m-hypergraph of size n with an associated prime of height q. We further prove that it is possible to construct connected m-hypergraphs under the same conditions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
