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Böttcher coordinates at wild superattracting fixed points

Hang Fu, Hongming Nie

Abstract

Let $p$ be a prime number, let $g(x)=x^{p^{2}}+p^{r+2}x^{p^{2}+1}$ with $r\in\mathbb{Z}_{\geq0}$, and let $φ(x)=x+O(x^{2})$ be the Böttcher coordinate satisfying $φ(g(x))=φ(x)^{p^{2}}$. Salerno and Silverman conjectured that the radius of convergence of $φ^{-1}(x)$ in $\mathbb{C}_{p}$ is $p^{-p^{-r}/(p-1)}$. In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result.

Böttcher coordinates at wild superattracting fixed points

Abstract

Let be a prime number, let with , and let be the Böttcher coordinate satisfying . Salerno and Silverman conjectured that the radius of convergence of in is . In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result.
Paper Structure (5 sections, 17 theorems, 80 equations)

This paper contains 5 sections, 17 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Some preliminary lemmas
  3. Proof of Theorem \ref{['thm1.2']}
  4. Proof of Theorem \ref{['thm1.3']}
  5. Proof of Theorem \ref{['thm1.7']}

Key Result

Theorem 1.2

Let $p$, $N$, $d$, and $c$ satisfy Condition condA or condB. Then the maximal convergent open disks of $\varphi_{c}(z)$ and $\varphi_{c}^{-1}(z)$ are both $D(\infty,r_{N}^{1/d})=\{z\in\mathbb{C}_{p}:|z|_{p}>r_{N}^{1/d}\}$, where Moreover, $\varphi_{c}(z)$ gives a bijective isometry from $D(\infty,r_{N}^{1/d})$ onto itself.

Theorems & Definitions (34)

  • Conjecture 1.1: Salerno and Silverman
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2: Legendre
  • ...and 24 more