\input{preamble.tex}
\unittitle{Discs and annuli}

In this unit, we 
introduce $p$-adic closed discs and annuli, but in a purely 
ring-theoretic fashion.
This avoids having to introduce any $p$-adic analytic geometry.

Throughout the unit (and in all later units, unless explicitly
contravened), let $K$ be a field complete for a nontrivial 
nonarchimedean valuation $|\cdot|$. Assume that $K$ has characteristic $0$, but
the residue field $\kappa_K$ has characteristic $p>0$. Also assume that things
are normalized so that $|p| = p^{-1}$.

\section{Power series on closed discs and annuli}

We start by introducing some rings that should be thought of as the analytic
functions on a closed disc $|t| \leq \beta$, or a closed annulus $\alpha
\leq |t| \leq \beta$. As noted in the introduction, this is more properly
done in a framework of $p$-adic analytic geometry, but we will avoid this
framework.

For $\alpha,\beta > 0$, put
\[
K \langle \alpha/t, t/\beta \rangle = 
\left\{\sum_{i \in \ZZ} c_i t^i \in K \llbracket t, t^{-1} \rrbracket:
\lim_{i \to \pm \infty} |c_i| \rho^i = 0 \quad (\rho \in [\alpha, \beta]).
 \right\}.
\]
That is, consider formal bidirectional power series which converge 
whenever you plug in a value for $t$ with $|t| \in [\alpha,\beta]$,
or in other words, when $\alpha/|t|$ and $|t|/\beta$ are both at most 1;
it suffices to check for $\rho = \alpha$ and $\rho = \beta$.
Although formal bidirectional power series do not form a ring, the
subset $K \langle \alpha/t, t/\beta \rangle$ does form a ring under
the expected operations.

If $\alpha = 0$, 
the only reasonable interpretation of the previous definition is to
require $c_i = 0$ for $i < 0$. When there are no negative powers of $t$,
it is redundant to require the convergence for $\rho < \beta$.
In other words,
\[
K \langle 0/t, t/\beta \rangle = K \langle t/\beta \rangle
= \left\{\sum_{i=0}^\infty c_i t^i \in K \llbracket t \rrbracket:
\lim_{i \to \infty} |c_i| \beta^i = 0\right\}.
\]
One could also allow $\beta = \infty$ for a similar effect in the other
direction. More succinctly put, we identify
$K \langle \alpha/t,t/\beta \rangle$ with
$K \langle \beta^{-1}/t^{-1}, t^{-1}/\alpha^{-1} \rangle$.

\section{Gauss norms and Newton polygons}

The rings $K \langle \alpha/t,t/\beta \rangle$
quite a lot like polynomial rings (or Laurent polynomial rings,
in case $\alpha \neq 0$) in one variable. The
next few statements are all instances of this analogy. 

From the definition of $K \langle \alpha/t, t/\beta \rangle$,
we see that it carries a well-defined
$\rho$-Gauss norm 
\[
\left| \sum_i c_i t^i \right|_\rho = \max_i \{|c_i| \rho^i\}
\]
for any $\rho \in [\alpha, \beta]$. For $\rho = \alpha = 0$, this reduces
to simply $|c_0|$.

The additive version is this is to take $r \in [-\log \beta, -\log \alpha]$
and put
\[
v_r\left(\sum c_i t^i\right) = \min_i \{v(c_i) + ri\},
\]
where $v(c) = -\log |c|$. This is the same formula as we had for the
sloped valuation function on a polynomial ring, so we may repeat the proof
to obtain the following.
\begin{lemma}
For $r \in [-\log \beta, -\log \alpha]$, the function $v_r$
on $K \langle \alpha/t, t/\beta \rangle$ is a valuation;
in particular, $v_r(xy) = v_r(x) + v_r(y)$. Equivalently,
for $\rho \in [\alpha, \beta]$, the $\rho$-Gauss norm on 
$K \langle \alpha/t, t/\beta \rangle$ is really a norm; that is,
it indeed satisfies $|fg|_\rho = |f|_\rho |g|_\rho$.
\end{lemma}

One may define the \emph{Newton polygon} for an element $x = \sum x_i t^i
\in K \langle \alpha/t, t/\beta \rangle$ 
as the lower convex hull of the set
\[
\{(-i, v(x_i)): i \in \ZZ, x_i \neq 0\},
\]
except that we only keep the slopes in
$[-\log \beta, -\log \alpha]$. 
\begin{prop} \label{P:Hadamard plus}
Let $x = \sum_i x_i t^i \in K \langle \alpha/t, t/\beta \rangle$ be nonzero.
\begin{enumerate}
\item[(a)]
The Newton polygon of $x$ has finite width.
\item[(b)]
The function $r \mapsto v_r(x)$ on $[-\log \beta, -\log \alpha]$
is continuous, piecewise affine, and convex.
\item[(c)]
The function $\rho \mapsto |x|_\rho$ on $[\alpha,\beta]$ is continuous
and log-concave. The log-concavity means that
$\rho,\sigma \in [\alpha,\beta]$ 
and $c \in [0,1]$, put $\tau = \rho^c \sigma^{1-c}$; then
\[
|x|_{\tau} \leq |x|_\rho^c |x|_{\sigma}^{1-c}.
\]
\item[(d)]
If $\alpha = 0$, then $v_r$ is decreasing on $[-\log \beta, \infty)$;
in other words, for all $\rho \in [0, \beta]$,
$|x|_\rho \leq |x|_\beta$.
\end{enumerate}
\end{prop}
Part (c) should be thought of as a
nonarchimedean analogue of the Hadamard three circle theorem.
\begin{proof}
We have (a) because there is a least $i$ for which
$|c_i| \alpha^i$ is maximized, and there is a greatest $j$ for which
$|c_j| \beta^j$ is maximized. This implies (b) because as in the polynomial
case, we may interpret $v_r(x)$ as the $y$-intercept of the supporting line
of the Newton polygon of slope $r$. This in turn implies (c),
and (d) is a remark made earlier.
\end{proof}

When dealing with the ring $K \langle \alpha/t, t/\beta \rangle$, the following completeness property
will be extremely useful.
\begin{prop}\label{P:frechet}
The ring $K \langle \alpha/t, t/\beta \rangle$ is Fr\'echet complete for the norms $|\cdot|_\rho$
for all $\rho \in I$. That is, if $\{x_n\}_{n=0}^\infty$ is a sequence which is
simultaneously Cauchy under $|\cdot|_\rho$ for all $\rho \in I$, then 
it is convergent. (By Proposition~\ref{P:Hadamard plus},
it suffices to check the Cauchy property at each nonzero endpoint of $I$.)
\end{prop}
\begin{proof}
Exercise.
\end{proof}

For instance, the completeness property is used in the construction of
multiplicative inverses.
\begin{lemma} \label{L:unit}
If $\alpha = 0$ (resp.\ $\alpha > 0$),
a nonzero element $f \in K \langle \alpha/t, t/\beta \rangle$ is a 
unit if and only if $v_r$ is constant (resp.\ affine)
on $[-\log(\beta),-\log(\alpha)]$.
\end{lemma}
\begin{proof}
We will just consider the case $\alpha > 0$; the other case is similar 
(and easier).
Put $f = \sum_{i \in \ZZ} f_i t^i$. Note that the following are equivalent:
\begin{enumerate}
\item[(a)]
there is a single $i$ for which $|f|_\rho = |f_i| \rho^i$ for all $\rho \in 
[\alpha, \beta]$;
\item[(b)]
the function $r \mapsto v_r(f)$ on $[-\log(\beta), -\log(\alpha)]$
 is affine;
\item[(c)]
the Newton polygon of $f$ has no slopes in $[-\log(\beta), -\log(\alpha)]$.
\end{enumerate}
By (c), these conditions all hold if $f$ is a unit. Conversely, if these conditions hold,
then the series
\[
(f_it_i)^{-1} (1 - (f_it^i-f)/(f_i t^i))^{-1}
= \sum_{j=0}^\infty (f_it^i-f)^j (f_i t_i)^{-j-1}
\]
converges by Proposition~\ref{P:frechet}, and its limit is an inverse of $f$.
\end{proof}

\section{Factorization results}

\begin{prop}[Weierstrass preparation] \label{P:weierstrass}
Suppose that $f = \sum_{i \in \ZZ} f_i t^i \in K \langle \alpha/t, t/\beta \rangle$, and that $\rho \in [\alpha,\beta]$ 
is such that there is a unique $m \in \ZZ$ maximizing $|f_m| \rho^m$.
Then there is a unique factorization $f= f_m t^m g h$ with 
\begin{align*}
g &\in K \langle \alpha/t, t/\beta \rangle \cap K \llbracket t \rrbracket
= K \langle t/\beta \rangle, \\
h &\in K \langle \alpha/t, t/\beta \rangle \cap K \llbracket t^{-1} \rrbracket
= K \langle \alpha/t \rangle,
\end{align*}
$|g|_\rho = |g_0| = 1$, and $|h-1|_\rho < 1$.
\end{prop}
\begin{proof}
The master slope factorization applies thanks to
Property~\ref{P:frechet}.
\end{proof}

In light of the finite width property of the Newton polygon,
the following should not be a surprise.
\begin{prop}[More Weierstrass preparation] \label{P:pid}
For $f \in K \langle \alpha/t, t/\beta \rangle$, there exists a polynomial $P \in K[t]$ and a unit
$g \in K \langle \alpha/t, t/\beta \rangle^\times$ such that $f = Pg$. In particular, $K \langle \alpha/t, t/\beta \rangle$ is a
principal ideal domain.
\end{prop}
\begin{proof}
Using Proposition~\ref{P:weierstrass}, we may reduce to two instances
of the case $\alpha = 0$, so we restrict to that case hereafter. 
Put $f = \sum_i f_i t^i$, and choose $m$ maximizing
$|f_m| \beta^m$.
Let $R$ be the ring of formal sums $\sum_i c_i t^i$ of series
with $|c_i| \beta^i$ bounded as $i \to -\infty$ and tending to 0
as $i \to +\infty$.
Let $e$ be the inverse of $\sum_{i=0}^m f_i t^i$ in $R$,
and apply master slope factorization to 
factor $ef = gh$ in $R$, in which $g$ is a unit in
$K \langle t/\beta \rangle$
by Lemma~\ref{L:unit}. Now $h \sum_{i=0}^m f_i t^i = f g^{-1}$ belongs to
\[
K \llbracket t \rrbracket \cap t^m K \llbracket t^{-1} \rrbracket.
\]
It is thus a polynomial of degree $m$, proving the claim.
\end{proof}

We will make frequent and often implicit use of the following patching lemma.
\begin{lemma}[Patching lemma] \label{L:patching}
Suppose $\alpha \leq \gamma \leq \beta \leq \delta$.
Let $M_1$ be a finite free module over $K \langle \alpha/t, t/\beta \rangle$,
let $M_2$ be a finite free module over $K \langle \gamma/t, t/\delta \rangle$,
and suppose we are given an isomorphism
\[
\psi: M_1 \otimes K \langle \gamma/t,t/\beta \rangle \cong
M_2 \otimes K \langle \gamma/t,t/\beta \rangle.
\]
Then we can find a finite free module $M$ over $K \langle \alpha/t,
t/\delta \rangle$ and isomorphisms
$M_1 \cong M \otimes K \langle \alpha/t, t/\beta \rangle$,
$M_2 \cong M \otimes K \langle \gamma/t, t/\delta \rangle$
inducing $\psi$. Moreover, $M$ is determined by this
requirement up to unique isomorphism.
\end{lemma}
\begin{proof}
We will only explain the case $\alpha > 0$; the case $\alpha = 0$ is similar.

Choose bases of $M_1$ and $M_2$ and let
$A$ be the $n \times n$ matrix defining $\psi$;
then $A$ must be invertible over $K \langle \gamma/t,t/\beta \rangle$.
Choose $\rho \in [\gamma,\beta]$;
since $\det(A)$ is a unit in $K \langle \gamma/t,t/\beta \rangle$,
we can find an invertible $n \times n$ matrix $W$ over 
$K \langle \gamma/t,t/\beta \rangle$ such that $\det(WA) = 1$.
(For instance, take $W = \Diag(\det(A)^{-1},1,\dots,1)$.)

It is then possible (see exercises)
to find invertible matrices $U,V$ over $K[t,t^{-1}]$ such that
$|UWAV - I_n|_\rho < 1$. By changing the initial choices of bases, we can
force ourselves into the case $|A-I_n|_\rho < 1$.

By using the master slope factorization 
in the matrix ring over
$K \langle \gamma/t,t/\beta \rangle$,
we can split $A$ as a product of an invertible matrix over
$K \langle t/\beta \rangle$ and an invertible matrix over
$K \langle \gamma/t \rangle$. Using these to change basis in $M_1$
and $M_2$, respectively, we can put ourselves in the situation where
$A=I_n$, in which case we may identify the bases of $M_1$ and $M_2$. Take
$M$ to be the free module over $K \langle \alpha/t,
t/\delta \rangle$ with the same basis.
\end{proof}

\notes

The Hadamard three circles theorem (Proposition~\ref{P:Hadamard plus}(c)) 
is a special
case of the fact that the \emph{Shilov boundary} of the annulus 
$\alpha \leq |t| \leq \beta$ consists of the two circles
$|t| = \alpha$ and $|t| = \beta$. For much amplification of this remark,
including a full-blown theory of harmonic functions on Berkovich analytic
curves, see [Thu05]. For an alternate presentation, restricted to
the Berkovich projective line but otherwise more detailed, see [BR07].

The patching lemma (Lemma~\ref{L:patching}) 
is a special case of the glueing property of coherent sheaves on affinoid
rigid analytic spaces, i.e., the theorems of Kiehl and Tate
[BGR84, Theorems~8.2.1/1 and~9.4.2/3].
The factorization argument in the proof, however, is older still;
it is the nonarchimedean version of what is called a \emph{Birkhoff
factorization} over an archimedean field.

\exercises

\begin{enumerate}
\item
Prove Proposition~\ref{P:frechet}. (Hint: it may be easiest to first
construct the limit using a single $\rho \in [\alpha, \beta]$, 
then show that it must also work for the other $\rho$.)
\item
Let $R$ be the ring of formal power series over $K$ which converge for
$|t| < 1$. Prove that $R$ is not noetherian; this is why I avoided introducing
it.
\item
Suppose $K$ is complete for a discrete valuation. Prove that any element of
$\gotho_K \llbracket t \rrbracket \otimes_{\gotho_K} K$ (that is,
a power series with bounded coefficients) is equal to a polynomial in $t$ times
a unit. Then prove that this fails if $K$ is complete for a nondiscrete valuation.
\item
Let $A$ be an $n \times n$ matrix over $K \langle \rho/t,t/\rho \rangle$
such that $|\det(A) - 1|_\rho < 1$. Prove that there exist invertible
matrices $U,V$ over $K[t,t^{-1}]$ such that $|U^{-1} A V - I_n|_\rho < 1$.
(Hint: perform approximate Gaussian elimination. An analogous argument,
but in more complicated notation, is [Ked04, Lemma~6.2]. We will see a similar
result in the unit on numerical analysis.)
\end{enumerate}

\end{document}