Grafos funcionais e dinâmica de polinômios sobre corpos finitos

Abstract

In this work, we present a method to characterize the structure of functional graphs of mnice polynomials over finite fields. We begin by reviewing elementary concepts about finite fields and other essential algebraic elements. Next, we introduce fundamental arithmetic tools critical to our objective, with emphasis on m-nice polynomials, a class of polynomials upon which we establish our results regarding the structure of functional graphs. Following this, we proceed to introduce the concept of graphs and their classifications, focusing on functional graphs and trees. These structures form the foundation of our study on the dynamics of polynomials over finite fields. We conclude the work by presenting results that characterize in detail the structure of functional graphs of m-nice polynomials over finite fields.