Date of Graduation

Spring 2016

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Mark Rogers

Keywords

local rings, polynomials, graph theory, primes, abstract algebra

Subject Categories

Mathematics

Abstract

We say a function generates a cycle if its output returns the initial value for some number of successive applications of . In this thesis, we develop a class of polynomial functions for finite local rings and associated functions . We show that the zeros of one are precisely the fixed points of the other and that every ring element is either one of these fixed points or is in a cycle of fixed length equal to the order of 2 in the associated group of units. Particular emphasis is given to rings of integers modulo the square of a prime. The construction of these polynomial functions arose from exploring constructions of polynomials in a similar manner to that embodied in the recent work of Dr. Cameron Wickham and Dr. Mark Rogers and the development of π-polynomials in "Polynomials Inducing the Zero Function on Local Rings”.

Copyright

© Tristen Kirk Wentling

Open Access

Included in

Mathematics Commons

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