## Honors Program Theses and Projects

5-14-2019

Thesis

#### Abstract

For this project, we explore nite eld dynamics and the various patterns of cycles of elements that emerge from the manipulation of a function and eld. Given a function f : Fp 􀀀! Fp, we can create a directed graph with an edge from c to f(c) for all c 2 Fp. We especially consider polynomials of the form f(x) = xd + c and investigate how varying the values of d and c affect the cycles in a given nite eld, Fp. We analyze data to look for graphs that result in cycles of length p. We also identify functions whose graphs have the same structure. We prove three main theorems. The rst theorem states that for p prime, if d1d2 1(mod p 􀀀 1), then the functions f1(x) = xd1 + c and f2(x) = xd2 􀀀 c have isomorphic graphs for any c 2 Zp. The second theorem states that if p 3 (mod 4) and p+1 = 2k then there are exactly p+1 2 choices for c in which the graph of f(x) = xp􀀀2 +c has one cycle of length p. The third theorem states that for c 2 Zp if d = p 􀀀 2 and p + 1 = 4q where q is a prime, then there are p􀀀3 2 choices for c in which the graph of f(x) = xd + c has one cycle of length p.

Mathematics

#### Thesis Comittee

Dr. Irina Seceleanu, Committee Member

Dr. Uma Shama, Committee Member