Stanley conjecture in small embedding dimension

Abstract

We show that Stanley's conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals.

An Algorithm for Computing the Multigraded Hilbert Depth of a Module

“…A first step was done by Herzog, Vladoiu and Zheng in [15], where they introduced a method for computing the Stanley depth of a factor of a monomial ideal which was later developed into an effective algorithm by Rinaldo in [22]. Some remarkable results in the study of the Stanley depth in the multigraded case were also presented by Apel (see [1], [2]), Herzog et al (see [13], [14]) and Popescu et al (see [3], [21]).…”

Stanley depth of monomial ideals with small number of generators

“…The Stanley conjecture for S/I was proved for ≤ 5 and in other special cases, but it remains open in the general case. See for instance, [1,3,4,7,9,11]. Let I ⊂ S be a monomial ideal.…”

On Characteristic Poset and Stanley Decomposition

“…We denote by uK[Z] the K-subspace of I/J generated by all elements uv where v is a monomial in K A Stanley decomposition of I/J is a presentation of the Z n -graded K-vector space I/J as a finite direct sum of Stanley spaces D : I/J = we call I a Stanley ideal. The conjecture is widely discussed in recent years for example [2], [4], [6], [8], [9].…”