Combinatorial Reductions for the Stanley Depth of I and S/I

2017/02/02 by Keller, Mitchel T., Young, Stephen J.
#05E40 #06A07 #13C13 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.00781

Abstract

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal I and the Stanley depth of its compliment, S/I. Using these results we are able to prove that if S is a polynomial ring with at most 5 indeterminates and I is a square-free monomial ideal, then the Stanley depth of S/I is strictly larger than the Stanley depth of I. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.

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