On the Stanley Depth of Powers of Monomial Ideals

Mathematics

S.A. Seyed Fakhari¹ 

[1] School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran;

关键词: complete intersection;    cover ideal;    depth;    edge ideal;    integral closure;    polymatroidal ideal;    Stanley depth;    Stanley’s inequality;    symbolic power;   

DOI  :  10.3390/math7070607

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown