Y, \lambda]_c$ by $$\begin{aligned}
\|f\|_{\lambda, c} &=&  \max |f_\mu|_\lambda p^{c\cdot \mu}.\end{aligned}$$

It’s easy to see that $F[[Y]] \subset F[X; Y, \lambda]_c \subseteq F[X; Y, \lambda]_{c'}$ if $c_i' \leq c_i$ for $i = 1, \cdots, n$.

The function $\| \; \|_{\lambda, c}$ is a non-Archimedean norm on $F[X; Y, \lambda]_c$.

On $F[[Y]]$, $\|\;\|_{\lambda, c}$ reduces to $| \; |_\lambda$. Suppose that $f$ and $g$ are elements of $F[X; Y,\lambda]_c$. Then $$\begin{aligned}
 \|f + g\|_{\lambda, c} &=& \max_{\mu}\{ |f_\mu+g_\mu|_\lambda p^{c\cdot \mu}\}\\
&\leq& \max_{\mu}\{\max\{|f_\mu|_\lambda, |g_\mu|_\lambda\}p^{c\cdot \mu}\} \\
&\leq& \max\{ \|f\|_{\lambda, c}, \|g\|_{\lambda, c}\}.\end{aligned}$$ Next, $$\begin{aligned}
 \|fg\|_{\lambda, c} &=& \max_{\sigma}\{ \left|\left(\sum_{\mu + \nu = \sigma} f_\mu g_\nu\right) \right|_\lambda p^{c\cdot \sigma} \} \\
&\leq& \max_{\sigma}\{ \max_{\mu + \nu = \sigma}\{ |f_\mu g_\nu|_\lambda\} p^{c\cdot \sigma} \} \\
&\leq& \max_{\mu, \nu}\{ |f_\mu|_\lambda |g_\nu|_\lambda p^{c\cdot ( \mu + \nu)} \} \\
&=& \|f\|_{\lambda, c}\|g\|_{\lambda, c}.\end{aligned}$$

$\displaystyle F[X; Y, \lambda] = \bigcup_{c \in {\mathbf{R}}^n, c_i> 0} F[X; Y, \lambda]_c$

Suppose that $f$ is an element of $F[X; Y, \lambda]$, then $|f_\mu|_\lambda p^{(C_f, \cdots, C_f)\cdot \mu}$ converges to $0$ as $|\mu|$ approaches $\infty$. Conversely, if $f$ is an element of $F[X; Y, \lambda]_c$, let $C_f = \min_{i} c_i$.

Suppose $f$ is an element of $F[X; Y,\lambda]$. Then $f(X,Y)$ is invertible 