## Mathematics, Department of

#### First Advisor

Thomas Marley

#### Date of this Version

Spring 5-2011

#### Abstract

The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and Yoshida comparing the Hilbert-Kunz multiplicity and the length of the ideal and provide a proof in the graded case.

Adviser: Thomas Marley

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Thomas Marley. Lincoln, Nebraska: May, 2011

Copyright 2011 Lori McDonnell