Date of Graduation
Master of Science in Mathematics
local rings, polynomials, graph theory, primes, abstract algebra
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”.
© Tristen Kirk Wentling
Wentling, Tristen Kirk, "A Cycle Generating Function On Finite Local Rings" (2016). MSU Graduate Theses. 1660.