Title of Thesis

Computing the Stanley depth

Author(s)

Muhammad Imran Qureshi

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[σ1, . . . , σn] 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 Ω1∩Ω2, S/(Ω1∩Ω2∩ Ω3), (Ωi)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 = (σ1, . . . , σt) ∩ (σt+1, . . . , σn) in S. In Chapter 3 we gave an upper bound for the Stanley depth of the ideal I′ = (I, σn+1, . . . , σn+p) ⊂ S′ = S[σn+1, . . . , σn+p]. We gave similar upper bounds for the Stanley depth of the ideal (In;2, σn+1, . . . , σn+p), where In;2 is the square free Veronese ideal of degree 2 in n variables. These results are from our paper [11].

1

1.1 Graded rings and modules
1.2 Monomial ideals
1.3 Primary decomposition of modules
1.4 Depth and Dimension
1.5 Stanley decompositions, Stanley depth and Stanley’s Conjecture

2.1 A lower bound for Stanley’s depth of some cycle modules
2.2 An upper bound for Stanley’s depth of some cycle modules
2.3 An illustration
2.4 A lower bound for Stanley’s depth of some ideals
2.5 Applications

3.1 Upper bounds for the Stanley depth of squarefree monomial ideal when some variables are added
3.2 Upper bounds for the Stanley depth of squarefree Veronese ideal when some variables are added
3.3 Comparison of bounds