Brian Rice

Harvey Mudd College Mathematics 2008

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Thesis Information

Thesis Proposal: Proposal: Rigid Divisibility Sequences
Final Report: Rigid Divisibility Sequences Generated by Polynomial Iteration
Thesis Advisor: Prof. Nicholas Pippenger
Second Reader: Prof. Christopher Towse

Rigid Divisibility Sequences

A rigid divisibility sequence is a sequence {an} of algebraic integers (contained in the ring OK of integers of a number field K) satisfying the following property: for every prime ideal P of OK, there is an exponent d such that whenever an is contained in P, then an is contained in Pd but not in Pd+1. For a polynomial f(x) with coefficients in OK, we consider the sequence given by a1 = f(0) and an+1 = f(an) for n > 0. The goal of this project is to characterize as fully as possible those polynomials f(x) which generate rigid divisibility sequences in this way.

Non-Thesis Research

From Summer 2006: Primitive Prime Divisors in Polynomial Arithmetic Dynamics. This paper is the result of work done at the 2006 REU in Number Theory at University of Wisconsin-Madison, under supervision of Prof. Ken Ono. It is published in the Integers Journal (2007).
From Summer 2007: On-line Distributed Traffic Grooming, with R. Jordan Crouser (Smith), Adrian Sampson (HMC), and Ran Libeskind-Hadas (HMC). This paper came out of research at the 2007 REU in Computer Science at Harvey Mudd College, under supervision of Prof. Ran Libeskind-Hadas. It has just been accepted for the IEEE ICC '08.