Computing the Stanley depth

Abstract

In the first Chapter some definitions and necessary results from commutative algebra are given. Also some detail are given which leads us towards the Stanley decompositions of multigraded S-modules, where S = F [σ 1 , . . . , σ n ] is a polynomial ring in n variables over a field F . In the end of the chapter we give some important results about the Stanley decompositions and Stanley’s conjecture. In Chapter 2 for the given monomial primary ideals Ω and Ω ′ of S, we gave an upper bound for the Stanley depth of S/(Ω ∩ Ω ′ ) which is reached if Ω, Ω ′ are irreducible. Also we showed that Stanley’s Conjecture holds for Ω 1 ∩Ω 2 , S/(Ω 1 ∩Ω 2 ∩ Ω 3 ), (Ω i ) i being some irreducible monomial ideals of S. These results are published in our paper [23]. For integers 1 ≤ t < n consider the ideal I = (σ 1 , . . . , σ t ) ∩ (σ t+1 , . . . , σ n ) in S. In Chapter 3 we gave an upper bound for the Stanley depth of the ideal I ′ = (I, σ n+1 , . . . , σ n+p ) ⊂ S ′ = S[σ n+1 , . . . , σ n+p ]. We gave similar upper bounds for the Stanley depth of the ideal (I n,2 , σ n+1 , . . . , σ n+p ), where I n,2 is the square free Veronese ideal of degree 2 in n variables. These results are from our paper [11].