# Modulus of growth in $p$-adic spherically complete field of $\mathbb C_p$

Let $$F$$ be the spherically complete extension of $$\mathbb C_p$$ and $$(a_n)_{n\in\mathbb N}$$ be a sequence of $$\mathbb C_p$$ such that for all $$r\in\mathbb R$$, one has $$\lim_{n\to+\infty}|a_n|_pr^n=0.$$ Put $$f(z)=\sum_{n\ge0}a_nz^n$$. For a non-negative real number $$r$$ define the modulus of growth in $$F$$ by $$M_F(f,r)=\inf\{v_p(f(z))\mid z\in F\text{ with }v_p(z)\ge r\}.$$ For a non-negative rational number $$r$$ define the modulus of growth in $$\mathbb C_p$$ by $$M_{\mathbb C_p}(f,r)=\inf\{v_p(f(z))\mid z\in\mathbb C_p\text{ with }v_p(z)\ge r\}.$$ My question: for every positive real number $$r$$, can one find a sequence $$(r_n)_{n\in\mathbb N}$$ of rational numbers such that $$M_F(f,r)=\lim_{n\to+\infty}M_{\mathbb C_p}(f,r_n)?$$

• For $f$ a $p$-adic Laurent series and $r$ in the open disk of convergence, reducing $p^m f(z)$ modulo $(p^l,z^N)$ don't you get that $\inf_{v(z) \ge r} v(f(z)) = \inf_n v(a_n) + rn$ Oct 25, 2019 at 20:31

It is known that for an entire function $$f(x)=\sum_{n\geq 0} a_nx^n\in{\mathbb C}_p[[x]]$$, the function $$\begin{equation*} r\longrightarrow M(f,r)=\inf_{n\geq 0} v(a_n)+n r=\inf_{v(x)x\geq r} v(f(x)) \end{equation*}$$ is continuous and piecewise affine, cf. A. Robert,\emph{A course in $$p$$-adic analysis}, Springer Verlag GTM 198.