Bounds of Stanley depth

We answer positively a question of Asia Rauf for the case of intersections of three prime ideals generated by disjoint sets of variables and we present several inequalities on Stanley depth. This is a detailed presentation of our talk at the conference on ”Fundamental structures of algebra” in honor of Prof. Serban Basarab at his 70-th anniversary. Let S = K[x1, . . . , xn] be a polynomial algebra over a field K, I ⊂ J ⊂ S two monomial ideals and M = J/I. The depth of M is a homological invariant and depends on the characteristic of the field K. For example if I is the Stanley-Reisner ideal associated to the triangulation of the projective real plane PR then depthS/I = 3 if and only if the characteristic of K is not 2, otherwise depth S/I = 2 (see [16]). This is because the singular homology H̃1(PR;K) = 0 if and only if the characteristic of K is not 2, otherwise H̃1(PR;K) = K. In 1982 Stanley [18] introduces a new invariant the so-called the Stanley depth, which is combinatorially defined and so does not depend on the characteristic of the field K. Given a monomial u ∈ (J \ I) and Z ⊂ {x1, . . . , xn}, we say that ûK[Z], û = u + J , is a Stanley space of dimension |Z| if it is free over K[Z]. A Stanley decomposition of J/I is a finite direct sum of Stanley spaces, D : J/I = ⊕i=1uiK[Zi], and we call sdepth D = min{|Zi|} the Stanley depth of D . For example the Stanley decomposition D : K[x, y]/(x, xy) = K[y]⊕xK has sdepth D = 0. We define sdepthS J/I = max{sdepth D : D Stanley decomposition of J/I}. There exists an infinite set of Stanley decompositions and apparently it is impossible to find sdepth in general. Herzog-Vladoiu-Zheng [5] reduced the problem to find a partition of a finite ordered set. Stanley conjectured that