Integrality properties of B\"ottcher coordinates for one-dimensional superattracting germs

TL;DR

This paper investigates the integrality of coefficients in Böttcher coordinates for one-dimensional superattracting germs, revealing conditions under which these coefficients are integral and exploring their properties in $p$-adic contexts.

Contribution

It establishes new integrality results for Böttcher coordinate coefficients in various settings, including $p$-adic rings and specific polynomial forms, with implications for $p$-adic dynamics.

Findings

If $R=\mathbb{Z}_p$ and $\varphi(x)\in x^p + px^{p+1}R[[x]]$, then $f_\varphi(x)\in R[[x]]$.

For $\varphi(x)\in x^m + mx^{m+1}R[[x]]$, the coefficients of $f_\varphi(x)$ are in $R$ and have a factorial-based form.

When $m=p^2$, certain coefficients satisfy specific congruences modulo $p$.

Abstract

Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let $m\ge2$. The B\"ottcher coordinate of a power series $\varphi(x)\in x^m + x^{m+1}R[\![x]\!]$ is the unique power series $f_\varphi(x)\in x+x^2K[\![x]\!]$ satisfying $\varphi\circ f_\varphi(x) = f_\varphi(x^m)$. In this paper we study the integrality properties of the coefficients of $f_\varphi(x)$, partly for their intrinsic interest and partly for potential applications to $p$-adic dynamics. Results include: (1) If $p$ is prime and $R=\mathbb Z_p$ and $\varphi(x)\in x^p + px^{p+1}R[\![x]\!]$, then $f_\varphi(x)\in R[\![x]\!]$. (2) If $\varphi(x)\in x^m + mx^{m+1}R[\![x]\!]$, then $f_\varphi(x)=x\sum_{k=0}^\infty a_kx^k/k!$ with all $a_k\in R$. (3) In (2), if $m=p^2$, then $a_k\equiv-1\pmod{p}$ for all $k$ that are powers of $p$.