The inverse stability of Artin-Schreier polynomials over finite fields
Kaimin Cheng
TL;DR
This paper addresses inverse stability of Artin–Schreier polynomials over finite fields, focusing on polynomials of the form $g(X)=X^{p}-X+\xi$ with ${\rm Tr}_{{\mathbb F}_{q}}(\xi)\ne 0$. It reduces the question to the denominators $d_{n,g}(X)$ of the iterates of $G(X)=1/g(X)$ and derives a precise criterion: $g$ is inversely stable over ${\mathbb F}_{q}$ if and only if ${\rm Tr}_{{\mathbb F}_{q}}(a_n c_n^{-1})\ne 0$ for every $n\ge 1$, where $(a_n)$, $(c_n)$ are defined recursively from $\xi$. The sufficiency part uses a chain of field extensions and irreducibility of auxiliary polynomials to show each $d_{n,g}(X)$ has degree $p^{n}$ and is irreducible, hence all $d_{n,g}(X)$ are distinct and irreducible. The results yield a practical trace-based test for inverse stability in the key case $e>t=1$, and the methodology suggests extensions to other Artin–Schreier-type cases, with periodicity offering finite checks in some instances. Overall, the work resolves the $e>t=1$ case of the inverse stability problem and provides a framework for broader parameter regimes.
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 $\mathbb{F}_q$.