Dynamics of polynomial maps over finite fields

TL;DR

This paper investigates the structure of functional graphs generated by polynomial maps over finite fields, providing complete characterizations for certain classes including monomial maps, and deriving properties like connected components and fixed points.

Contribution

It offers a complete description of the functional graphs for specific polynomial maps over finite fields, including monomials, under certain regularity conditions.

Findings

Characterization of the functional graph structure for maps satisfying regularity conditions

Explicit formulas for the number of connected components and fixed points

Complete description of the graphs for monomial maps

Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $n$ be a positive integer. In this paper, we study the digraph associated to the map $x\mapsto x^n h(x^{\frac{q-1}{m}})$, where $h(x)\in\mathbb{F}_q[x].$ We completely determine the associated functional graph of maps that satisfy a certain condition of regularity. In particular, we provide the functional graphs associated to monomial maps. As a consequence of our results, the number of connected components, length of the cycles and number of fixed points of these class of maps are provided.