On the functional graph of $f(X)=c(X^{q+1}+aX^2)$ over quadratic extensions of finite fields

TL;DR

This paper thoroughly analyzes the functional graph structure of a specific polynomial map over quadratic extensions of finite fields, revealing detailed dynamics for certain parameter choices and partial results for others.

Contribution

It provides a complete description of the dynamics of the map for specific parameters and partial insights for more general cases over finite field extensions.

Findings

Complete dynamics description for a=±1 and c∈F_q^*

Partial results for a∈F_q^* extbackslash extbackslash{±1}

Insights into the structure of the functional graph over quadratic extensions

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements and $char(\mathbb{F}_q)$ odd. In this article we will describe completely the dynamics of the map $f(X)=c(X^{q+1}+aX^2)$, for $a=\{\pm1\}$ and $c\in\mathbb{F}_q^*$, over the finite field $\mathbb{F}_{q^2}$, and give some partial results for $a\in\mathbb{F}_q^*\setminus\{\pm1\}$.