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On the dynamical Galois group of certain affine polynomials in positive characteristic

Andrea Ferraguti, Guido Maria Lido

TL;DR

The paper determines when the dynamical Galois group $G_\infty(f,0)$ of polynomials $f(x)=x^q+t x+s$ in characteristic $p$ attains its maximal affine-Carlitz size by combining Hayes' explicit class field theory with Artin–Schreier techniques. It introduces a universal affine Carlitz polynomial $\tilde{f}=x^q+tx+s$ and proves $G_\infty(\tilde{f},0)\cong \Phi_\infty$, then uses specialization to $f(x)=x^q+t_0x+s_0$ to derive concrete surjectivity criteria for the natural embedding $G_\infty(f,0)\hookrightarrow \Phi_\infty$. The main result provides equivalent conditions: (i) $-t_0$ is not an $\ell$-th power for all $\ell| (p-1)$, (ii) a specified set of elements is $\mathbb{F}_p$-linearly independent in $\mathcal{V}(K)$, and (iii) $f$ is irreducible; and when a $\theta_0$ with $\theta_0^{q-1}=-t_0$ exists, a refined independence condition characterizes surjectivity onto $\Phi_{\infty,1}$. The work extends Stoll’s quadratic criterion to positive characteristic via a deep mix of function-field class field theory and dynamical Galois theory, with implications for the arithmetic of iterates and their discriminants.

Abstract

We use explicit class field theory of rational function fields to prove a dynamical criterion for a polynomial of the form $x^{p^r}+ax+b$ over a field of characteristic $p$ to have dynamical Galois group as large as possible. When $p=2$ and $r=1$ this yields an analogue in characteristic $2$ of the celebrated criterion of Stoll for quadratic polynomials over fields of characteristic not $2$.

On the dynamical Galois group of certain affine polynomials in positive characteristic

TL;DR

The paper determines when the dynamical Galois group of polynomials in characteristic attains its maximal affine-Carlitz size by combining Hayes' explicit class field theory with Artin–Schreier techniques. It introduces a universal affine Carlitz polynomial and proves , then uses specialization to to derive concrete surjectivity criteria for the natural embedding . The main result provides equivalent conditions: (i) is not an -th power for all , (ii) a specified set of elements is -linearly independent in , and (iii) is irreducible; and when a with exists, a refined independence condition characterizes surjectivity onto . The work extends Stoll’s quadratic criterion to positive characteristic via a deep mix of function-field class field theory and dynamical Galois theory, with implications for the arithmetic of iterates and their discriminants.

Abstract

We use explicit class field theory of rational function fields to prove a dynamical criterion for a polynomial of the form over a field of characteristic to have dynamical Galois group as large as possible. When and this yields an analogue in characteristic of the celebrated criterion of Stoll for quadratic polynomials over fields of characteristic not .
Paper Structure (6 sections, 9 theorems, 38 equations)

This paper contains 6 sections, 9 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Artin-Schreier theory
  4. The class field theory of
  5. Dynamical Galois groups of affine Carlitz polynomials
  6. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $K$ be a field of characteristic not $2$ and let $f=x^2+c\in K[x]$ be such that $c_n\ne 0$ for every $n\ge 1$. Let $\Omega_\infty$ be the full automorphism group of the infinite, rooted, regular, binary tree. Then $G_\infty(f)=\Omega_\infty$ if and only if the adjusted post-critical orbit $\{c_1

Theorems & Definitions (15)

  • Theorem 1.1: stoll
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Lemma 3.1
  • ...and 5 more