References: p-adic Differential Equations by Kiran S. Kedlaya
Let \(K\) be a complete field of characteristic \(0\) for a norm \(|\cdot |\) with residue field \(\kappa\) of characteristic \(p\) (which is allowed to be zero as well o.o). The norm is normalized \(|p| = p^{-1}\) (if \(p\neq 0\)).
Recall in rigid analytic geometry of \(\mathbb{P}^1\), the most important examples of affinoid subset \(F\subset \mathbb{P}^1\) are the ring domain \(\{|z|=1\}\) and the thick ring domain \(\{r_1\le |z|\le r_2\}\). The ring of functions on these domains are given by power series rings with convergence conditions, similar to a Tate algebra.
The classical Tate algebra is the ring of power series \(\sum_{n\ge 0} a_n t^n\) with \(|a_n|\xrightarrow{n\to \infty} 0\), corresponding to \(\mathcal{O}(\{|z|\le 1\})\).
For \(F=\{|z|\le |\beta|\}\), this is just a slight generalization of the Tate algebra, where you can also conveniently think of the Tate algebra of the variable \(z/\beta\) instead of \(z\).
\[\begin{align*} \mathcal{O}(F) &= K\left\langle \frac{z}{\beta} \right\rangle \\ &= \left\{ \sum_{n\ge 0} a_n \beta^{-n} t^n : |a_n|=o(1) \right\} \\ &= \left\{ \sum_{n\ge 0} c_n t^n : |c_n\beta^n|=o(1) \right\} \\ &= \left\{ \sum_{n\ge 0} c_n t^n : v(c_n)-n\cdot v(1/\beta)\xrightarrow{n\to \infty} \infty \right\} \end{align*}\]
\(F=\{|\alpha|\le |z|\}\). If we do a variable substitution and consider power series ring based at infinity, i.e. with variable \(\frac{\alpha}{z}\), the Tate algebra of variable \(\frac{\alpha}{z}\) give the ring \(\mathcal{O}(F)\).
\[\begin{align*} \mathcal{O}(F) &= K\left\langle \frac{\alpha}{z} \right\rangle \\ &= \left\{ \sum_{n\ge 0} a_n \alpha^n z^{-n}: |a_n|=o(1) \right\} \\ &= \left\{ \sum_{n\ge 0} c_n z^{-n}: |c_n \alpha^{-n}|=o(1) \right\} \\ &= \left\{ \sum_{n\ge 0} c_n z^{-n} : v(c_n)-(-n)\cdot v(1/\alpha)\xrightarrow{n\to \infty} \infty \right\} \end{align*}\]
For \(F=\{|\alpha|\le |z|\le |\beta|\}\), the thick ring domain. This \(\mathcal{O}(F)\) is in fact the intersection of \(K\langle \frac{z}{\beta} \rangle\) and \(K\langle \frac{\alpha}{z} \rangle\).
\[\mathcal{O}(F) =K\left\langle \frac{\alpha}{z}, \frac{z}{\beta} \right\rangle =\left\{ \sum_{n\in \mathbb{Z}} a_n t^n : \quad \begin{align*} &v(a_k)-kv(1/\beta)\xrightarrow{k\to \infty} \infty \\ &v(a_k)-kv(1/\alpha)\xrightarrow{k\to -\infty} \infty \end{align*} \right\} \]
or equivalently as
\[K\left\langle \frac{\alpha}{z}, \frac{z}{\beta} \right\rangle =\left\{ \sum_{n\in \mathbb{Z}} a_n t^n : \quad \begin{align*} &|a_k\alpha^k|=o(1) \\ &|a_k\beta^{k}|=o(1) \end{align*} \right\}. \]
Which pictorially is a V-shaped region, your function need to eventually go above the V-shape and the height to the V-shape needs to go to infinity on both sides. We also conveniently denote this ring as \(K\langle \frac{\alpha}{z}, \frac{z}{\beta} \rangle\).
For \(F=\{|z|=1\}\), the ring domain, we have
\[\begin{align*} \mathcal{O}(F) &= K\langle z, z^{-1}\rangle \\ &= \left\{ \sum_{n\in \mathbb{Z}} a_n t^n : |a_n|\xrightarrow{|n|\to \infty} 0 \right\} \\ &= \left\{ \sum_{n\in \mathbb{Z}} a_n t^n : v(a_n)-n\cdot v(1/1)\xrightarrow{|n|\to \infty} \infty \right\} \end{align*}\]
We define the ring with finite Gauss norm as
\[K[[t/\beta]]_0 := \left\{ \sum_{i=0}^\infty a_i t^i : \sup |a_i|\beta^i < \infty \right\}. \]
Clearly we have for \(\delta<\beta\) that
\[K\langle t/\beta \rangle \subset K[[t/\beta]]_0 \subset K\langle t/\delta \rangle\]
Note that we can also write it as
\[K[[t]]_0 = \mathfrak{o}_K[[t]]\otimes_{\mathfrak{o}_K} K\]
where \(\mathfrak{o}_K\) is the ring of integers of \(K\).
It is also possible to mix the two type of boundaries, producing rings like
\[K\langle \frac{\alpha}{z}, \frac{z}{\beta} ]]_0 =\left\{ \sum_{n\in \mathbb{Z}} a_n t^n : \quad \begin{align*} &\sup |a_k|\beta^k < \infty \\ &|a_k|\alpha^{k} = o(1) \end{align*} \right\} \]
\[K[[\frac{\alpha}{z}, \frac{z}{\beta} ]]_0 =\left\{ \sum_{n\in \mathbb{Z}} a_n t^n : \quad \begin{align*} &\sup |a_k|\beta^k < \infty \\ &\sup |a_k|\alpha^{k} < \infty \end{align*} \right\} \]
For which we have the simple property, for \(\delta\in [\alpha, \beta)\)
\[ K\langle \alpha/t, t/\beta\rangle \subset K\langle \alpha/t, t/\beta]]_0 \subset K\langle \alpha/t, t/\delta\rangle \]
Given an element \(x\) of \(K\langle \alpha/t, t/\beta\rangle\), we can associate a Newton polygon to it.
The full Newton Polygon \(\operatorname{NP}(x)\) is the lower convex hull of the set of points \((i, v(a_i))\).
The essential Newton Polygon \(\operatorname{ENP}(x) = \operatorname{NP}_{[\log |\alpha|, \log |\beta|]}(x)\) is generated by the subset of slopes within \([\log |\alpha|, \log |\beta|]\).
Denote by \(d_\lambda(f) = \inf (v(a_i)-\lambda i)\) the distance of the Newton polygon to the line \(y=\lambda x\).
Note: all the \(\log\) are respect to the base of the valuation. \(\log |\alpha|\) can also be more easily understood as \(v(1/\alpha)\).
\(\operatorname{ENP}(x)\) has finite length.
\(d_{(\cdot)}(x)\) is continuous and piecewise linear on \([\log |\alpha|, \log |\beta|]\).
\(|x|_{\lambda}\) is continuous on \([\log |\alpha|, \log |\beta|]\) and log-convex, for \(t\in [0,1]\) we have
\[|x|_{t\lambda+(1-t)\mu} \le |x|_{\lambda}^t |x|_{\mu}^{1-t}.\]
Or equivalently, for \(x\in K\langle \alpha/t, t/\beta\rangle\) we have that \(d_{(\cdot)}(x)\) is concave, i.e.
\[d_{t\lambda+(1-t)\mu}(x) \ge t d_{\lambda}(x) + (1-t) d_{\mu}(x).\]
Proof:
\[\begin{align*} \inf\left[v(a_i)-(t\lambda+(1-t)\mu)i\right] &=\inf\left[ t(v(a_i)-\lambda i) + (1-t)(v(a_i)-\mu i) \right] \\ &\ge t\inf(v(a_i)-\lambda i) + (1-t)\inf(v(a_i)-\mu i) \end{align*}\]
Taking \(|x|_\lambda = p^{-d_\lambda(x)}\) we obtain the log-convexity.
Since \(|x|_\lambda\) is log-convex, there exists a point \(\lambda_0\) (if we allow \(\pm \infty\)) such that \(|x|_{\lambda_0}\) is minimal, \(|x|_{\lambda}\) is decreasing for \(\lambda<\lambda_0\) and increasing for \(\lambda>\lambda_0\).
If \(\alpha=0\), this \(\lambda_0=-\infty\) and we have \(|x|_{\lambda}\) is increasing on \((-\infty, \log |\beta|]\).
The rings \(K\langle \alpha/t, t/\beta\rangle\) and \(K\langle \alpha/t, t/\beta]]_0\) are Frechet complete with respect to any Gauss norm \(|\cdot|_\lambda\) with slope \(\lambda\in [\log |\alpha|, \log |\beta|]\).
A non-zero element \(f\in K\langle t/\beta\rangle\) is a unit iff \(|f-c|_{\log |\beta|}<|f|_{\log |\beta|}\) for some unit \(c\in K^\times\).
A nonzero element \(f\in K\langle \alpha/t, t/\beta\rangle\) is a unit iff there exists a term \(ct^i\) such that \(|f-ct^i|_{\lambda}<|f|_{\lambda}\) for \(\lambda \in \{\log |\alpha|, \log |\beta|\}\).
A nonzero element \(f\in K\langle \alpha/t, t/\beta]]_0\) is a unit iff there exists a term \(ct^i\) such that \(|f-ct^i|_{\lambda}<|f|_{\lambda}\) for \(\lambda \in [\log |\alpha|, \log |\beta|)\).
In \(K\langle\alpha/t,t/\beta\rangle\) or \(K\langle\alpha/t,t/\beta]]_0\), if the series \(x\) has a unique term maximizing \(|x_m|\rho^m\) for some \(\rho\in [\alpha,\beta]\) or the supremum of \(|x_i|\rho^i\) is achievable with least achieving index \(m\), then by moving the term \(x_mt^m\) to center we can obtain the following factorization:
\[ x = x_mt^m \cdot g(t) \cdot h(t^{-1}) \]
with \(h(t^{-1})\in K\langle \alpha/t \rangle\Leftrightarrow h\in K\langle t/\alpha^{-1}\rangle\), \(|h(t^{-1})-1|<1\) and \(g(t)\in K\langle t/\beta \rangle\) or \(K\langle t/\beta ]]_0\) with \(|g|_{\log |\beta|}=1\) (If you are getting unique index \(m\), \(|g|_{\log |\beta|}<1\)).
Sitting between \(K\langle t/\beta\rangle \subset K[[t/\beta]]_0\), the set of elements in \(K[[t/\beta]]_0\) that comes from limit of rational functions \(K(t)\) is called the set of analytic elements, denoted \(K[[t/\beta]]_\operatorname{an}\).