## Mathematics, Department of

## First Advisor

Jack Jeffries

## Committee Members

Alexandra Seceleanu, Mark Walker, Arman Roohi

## Date of this Version

8-2024

## Document Type

Dissertation

## Citation

A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfilment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Jack Jeffries

Lincoln, Nebraska, August 2024

## Abstract

Say I hand you a shape, any shape. It could be a line, it could be a crinkled sheet, it could even be a the intersection of a cone with a 6-dimensional hypersurface embedded in a 7-dimensional space. Your job is to tell me about the pointy bits. This task is easier when you can draw the shape; you can you just point at them. When things get more complicated, we need a bigger hammer.

In a sense, that “bigger hammer” is what the ring of differential operators is to an algebraist. Then we will say some things and stuff that explains what we mean. At the end of the day, what we want to tell the reader is the following:

Differential operator theory in commutative algebra is about gaining the leverage of a partial derivative on polynomial expressions through a purely algebraic definition

An interesting result in the field is Bernstein’s inequality. It allows us to put bounds on dimension. This in turn allows us to gain further insight into complicated structures. In a sense, by placing big complicated structures (horrendously nonfinitely generated modules) next to an even bigger object (the ring of differential operators), the big appears small by comparison.

A related invariant is the Bernstein-Sato Polynomial. In may be thought of as an analog to the power rule taught in the first week of a calculus course, but generalized to functions that do not look like the power of a variable. In the work presented here, we develop an even more generalized version of this invariant, called the sandwich Bernstein-Sato polynomial. Similar results regarding Bernstein’s inequality can be found for this invariant, which lead us to new settings where the inequality holds. In particular, we will show that that Bernstein’s inequality holds for the ring of differential operators on the Segre embedding of projective spaces.

Advisor: Jack Jeffries

## Comments

Copyright 2024, David Lieberman. Used by permission