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The purpose of this paper is to provide methods for determining theassociated primes of the square of the Alexander dual of the edge ideal for anm-hypergraph H. We prove a general method for detecting associated primes ofthe square of the Alexander dual of the edge ideal based on combinatorialconditions on the m-hypergraph. Also, we demonstrate a more efficientcombinatorial criterion for detecting the non-existence of non-minimalassociated primes. In investigating 3-hypergraphs, we prove a surprisingextension of the previously discovered results for 2-hypergraphs (simplegraphs). For 2-hypergraphs, associated primes of the square of the Alexanderdual 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, givenany integers m, q, and n with 3\leq m\leq q\leq n, we construct a m-hypergraphof size n with an associated prime of height q. We further prove that it ispossible to construct connected m-hypergraphs under the same conditions.

Associated Primes of the Square of the Alexander Dual of Hypergraphs