#### Year of Publication

2008

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Uwe Nagel

#### Abstract

Making use of algebraic and combinatorial techniques, we study two topics: the arithmetic degree of squarefree strongly stable ideals and the *h*-vectors of matroid complexes.

For a squarefree monomial ideal, *I*, the arithmetic degree of *I* is the number of facets of the simplicial complex which has *I* as its Stanley-Reisner ideal. We consider the case when *I* is squarefree strongly stable, in which case we give an exact formula for the arithmetic degree in terms of the minimal generators of *I* as well as a lower bound resembling that from the Multiplicity Conjecture. Using this, we can produce an upper bound on the number of minimal generators of any Cohen-Macaulay ideals with arbitrary codimension extending Dubreil’s theorem for codimension 2.

A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible *h*-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question and we give some partial results for higher dimensions. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid *h*-vectors are pure *O*-sequences. Finally, we completely characterize the Stanley-Reisner ideals of matroid complexes.

#### Recommended Citation

Stokes, Erik, "THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE STRONGLY STABLE IDEALS" (2008). *University of Kentucky Doctoral Dissertations*. 636.

https://uknowledge.uky.edu/gradschool_diss/636