On the Stanley Depth of Powers of Monomial Ideals

S. A. Seyed Fakhari
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S. A. Seyed Fakhari: School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran

Mathematics, 2019, vol. 7, issue 7, 1-17

Abstract: In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.

Keywords: complete intersection; cover ideal; depth; edge ideal; integral closure; polymatroidal ideal; Stanley depth; Stanley’s inequality; symbolic power (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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