Keywords (Extracted from title, table of contents and
abstract of thesis) Irreducible, Monomial, Commutative,
Algebra, Stanley, Computing, Depth, Bounds, Monomial |
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]. |