Coefficient-Level B\"ottcher Theory for Wild Superattracting Germs of Degree $p^e$

Abstract

Let $p$ be an odd prime, let $e\ge2$, and put $q=p^e$. We study the wild family \[ \varphi_{r,e}(x)=x^q+qp^r x^{q+1}=x^{p^e}+p^{r+e}x^{p^e+1} \qquad (r\ge0), \] and the inverse B\"ottcher coordinate $f_{r,e}(x)=x\sum_{k\ge0}a_k(r,e)x^k/k!$ characterized by \[ \varphi_{r,e}(f_{r,e}(x))=f_{r,e}(x^q). \] For the clean family, we prove a complete mod-$p$ digit-sum law in the special fiber $r=0$. For the higher fibers $r\ge1$, we prove a coefficient-level theorem consisting of a global digit-weight lower bound, a leading monomial theorem on divisible non-pure classes, a lag-$e$ pure-power recursion, and subadditivity of the induced digit weight. This yields the pure-power branch word \[ (B^{e-1}A)^{\lceil r/e\rceil}B^\infty \] and the radius formula \[ \rho(f_{r,e})=p^{-\theta_{r,e}},\qquad \theta_{r,e}=p^{-e\lceil r/e\rceil}\left(\frac{1}{p-1}+e\lceil r/e\rceil-r\right). \] We then prove a tail-stable extension. In the special fiber, $p$-divisible tails preserve the digit-sum law modulo $p$. In the higher fibers, tails satisfying $v_p(\vartheta_h)\ge\Lambda_{r,e}(h+1)+1$ lie beyond the clean-family initial $\Lambda_{r,e}$-graded term and therefore preserve the leading terms, the pure-power branch word, the valuation asymptotic, and the radius. For $e=2$, this recovers the Salerno--Silverman degree-$p^2$ family and the Fu--Nie radius statement for the inverse coordinate in that family.