The inverse stability of Artin-Schreier polynomials over finite fields

TL;DR

This paper characterizes when certain Artin-Schreier polynomials over finite fields are inversely stable, meaning their iterated denominators remain irreducible and distinct, by providing necessary and sufficient conditions.

Contribution

It offers a complete characterization of inverse stability for a class of Artin-Schreier polynomials over finite fields, extending understanding of their iterative properties.

Findings

Identifies conditions for inverse stability of specific Artin-Schreier polynomials.

Proves irreducibility and distinctness of denominators under these conditions.

Enhances understanding of polynomial iteration over finite fields.

Abstract

Let $p$ be a prime number and $q$ a power of $p$. Let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$ and a polynomial $\varphi(X)\in\mathbb{F}_q[X]$, let $d_{n,\varphi}(X)$ denote the denominator of the $n$th iterate of $\frac{1}{\varphi(X)}$. The polynomial $\varphi(X)$ is said to be inversely stable over $\mathbb{F}_q$ if all polynomials $d_{n,\varphi}(X)$ are irreducible polynomial over $\mathbb{F}_q$ and distinct. In this paper, we characterize a class of inversely stable polynomials over $\mathbb{F}_q$. More precisely, for $\varphi(X)=X^{p^t}+aX+b\in\mathbb{F}_q[X]$ with $t$ being a positive integer, we provide a sufficient and necessary condition for $\varphi(X)$ to be inversely stable over $\mathbb{F}_q$.