Böttcher coordinates at wild superattracting fixed points.

Let p$p$ be a prime number, let g(x)=xp2+pr+2xp2+1$g(x)=x^{p^{2}}+p^{r+2}x^{p^{2}+1}$ with r∈Z⩾0$r\in \mathbb {Z}_{\geqslant 0}$, and let ϕ(x)=x+O(x2)$\phi (x)=x+O(x^{2})$ be the Böttcher coordinate satisfying ϕ(g(x))=ϕ(x)p2$\phi (g(x))=\phi (x)^{p^{2}}$. Salerno and Silverman conjectured that the radius of convergence of ϕ−1(x)$\phi ^{-1}(x)$ in Cp$\mathbb {C}_{p}$ is p−p−r/(p−1)$p^{-p^{-r}/(p-1)}$. In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result. [ABSTRACT FROM AUTHOR]