# Integral element over p-adic power series

Let $p$ be a prime number. and $R[[X]]$ be the ring of formal series with coefficients in a $p$-adic field $R$. Let $\Lambda=\mathbb{Z}_p[[X]]$.

Question 1) Does there exist an explicit description of the integral closure $B$ of $\Lambda$ in $\mathbb{Q}_p[[X]]$ ?

Producing elements in $B$ but not in $\Lambda$ seems a little bit complicated. Any fraction $F(X)=P(X)/Q(X)$ with $P,Q \in \Lambda$, $X \nmid Q(X)$ can be seen as an element of $\mathbb{Q}_p[[X]]$ by expanding $Q(X)$ at 0. But actually, by Weierstrass preparation theorem, if $F \in B$ then $F$ is already in $\Lambda$. On the other hand, the $p$-adic exponential, defined by $$exp_p(X) = \sum_n \frac{1}{n!}X^n \in \mathbb{Q}_p[[X]]$$ is not in $\Lambda$, but is integral over $\Lambda$. Indeed, it is a $p$-th root of $exp_p(pX)$, which is in $\Lambda$, as the quantity $\frac{p^n}{n!}$ is $p$-integral for all integer $n \geq 0$.

I don't have any idea of what should $B$ look like, but as the set of coefficients of any element of $B$ satisfy polynomial relations with $\mathbb{Z}_p$-coefficients, my guess is that they should not grow too fast.

Question 2) Does a element of $B$ have a positive radius of convergence ? That is, if $F(X)=\sum_n c_n X^n \in B$, then does there exist an integer $r \geq 0$ such that the sequence $(p^{-rn}|c_n|_p)_n$ is bounded ?

Such an $F$ is an element in $\mathbb{Z}_p[[X/p^r]][\frac{1}{p}]$, and is even an element of $\mathbb{Z}_p[[X/p^r]]$, as $F$ is integral over $\mathbb{Z}_p[[X/p^r]]$. If it is true, it would reduce Question 1 to :

Question 1') What is the integral closure of $\Lambda$ in $\mathbb{Z}_p[[X/p^r]]$ ($r \geq 1$ fixed) ?

• Item (2) must follow from the implicit function theorem for analytic functions. See for instance thm 6.1.2 of the book by Krantz and Parks. The corresponding inverse function theorem, for analytic functions, is in Serre's "Lie groups and Lie algebras" (page 73, towards the end of chapter II) – Laurent Berger May 30 '18 at 7:31
• Thank you very much, indeed such an F satisfies P(X,F(X))=0, for some P(X,Y) with Zp-coefficients. In order to apply (utlrametric analogue of) theorem, it seems to require F(0)= 0 (that can easily be assumed) and dP/dY(0,0) non-zero. I can't see why this second assumption would not be a problem. – Sharpsel May 30 '18 at 13:31
• Better yet : see the answers to mathoverflow.net/questions/116269 – Laurent Berger Jun 1 '18 at 11:23
• Great, so if I'm not mistaken, Question 2) follows from analytic Artin approximation theorem (Artin, On the solutions of analytic equations, Invent. Math.,5, (1968), 277-291.). The argument goes as follows : Let such an F, and assume F(0)=0 (it isn't a problem as F-F(0) is still integral), et let P(X,Y) in Zp[[X]][Y] be of positive degree in Y such that P(X,F(X))=0. There are only finitely many G in Frac(Qp[[X]]) satisfying the polynomial equation P(X,G(X))=0, so it is also true for solutions in Qp[[X]]. Take an integer e such that all the solutions are distincts mod X^e. – Sharpsel Jun 2 '18 at 16:48
• By analytic Artin approximation theorem (see thm 2.1 iml.univ-mrs.fr/~rond/Artin_survey.pdf for instance), there exist a power series Fà in Qp[[X]] with positive radius of convergence that satisfies a) P(X,F0(X))=0 and b) F = F0 mod X^e. It follows that F=F0, and it proves that F also has a positive radius of convergence. – Sharpsel Jun 2 '18 at 16:51