#### Year of Publication

2012

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department/School/Program

Mathematics

#### First Advisor

Dr. Uwe Nagel

#### Abstract

Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial.

#### Recommended Citation

Moore, Dennis, "HILBERT POLYNOMIALS AND STRONGLY STABLE IDEALS" (2012). *Theses and Dissertations--Mathematics*. 2.

https://uknowledge.uky.edu/math_etds/2