Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let $mge2$. The Bottcher 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 $varphicirc f_varphi(x) = f_varphi(x^m)$. 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 $f_varphi(x)=xsum_{k=0}^infty a_kx^k/k!$ with all $a_kin R$. (3) In (2), if $m=p^2$, then $a_kequiv-1pmod{p}$ for all $k$ that are powers of $p$.
تحميل البحث