\omega$. Further, $\omega$ is distinguished of degree $s$.*

3.  (Automorphism Theorem) *Let $g(X,Y) = \sum_{\mu}g_\mu(Y)X^\mu$ be an element of $F[X; Y,\lambda]$ where $\mu = (\mu_1, \cdots, \mu_n)$ and $X^\mu = X_1^{\mu_1}\cdots X_n^{\mu_n}$. If $g_\mu(Y)$ is not divisible by $Y$ for some $\mu$, where $\mu_n > 0$, then there exists an automorphism $\sigma$ of $F[X; Y, \lambda]$ such that $\sigma(g)$ is $X_n$-distinguished.*

4.  *$F[X; Y, \lambda]$ is noetherian and factorial.*

The Euclidean algorithm is the key part of this theorem. Our proof of this algorithm follows Manin’s proof of the analogous result for power series rings as written in Lang [@Lang], except that we have to keep careful track of more delicate estimates that arise from the general growth function $\lambda(x)$. The other results are classical consequences of this algorithm, which are proved in this paper, but the techniques are essentially unchanged from techniques utilized in proofs of analogous results for power series rings as given in Bosch, etc. [@Bosch].

This topic is also motivated by a considerable body of work concerning “$k$-affinoid” algebras from non-Archimedean analysis. Let $k$ be a complete non-Archimedean valued field, with a non-trivial valuation, and define $T_n = k{\langle}X_1, \cdots, X_n {\rangle}$, Tate’s algebra, to be 