ScalingStacks

Proposition 2.51. [Weierstrass division] Let $\mathcal{T}_{n}$ be the $k$-Tate algebra of multiradius $\underline{r}=\underline{1}$, then

1. (1)

   Let $f\in\mathcal{T}_{n}$ be an distinguished element in $z_{n}$ of degree $d$, and $g\in\mathcal{T}_{n}$ be any element. Then there exist unique $r\in\mathcal{T}_{n-1}[z_{n}]$ of degree less than $d$ in $z_{n}$ and $q\in\mathcal{T}_{n}$ such that $g=q\cdot f+r$. Moreover $\vvvert g\vvvert_{\mathcal{T}_{n}}=\max\{\vvvert q\vvvert_{\mathcal{T}_{n}},\vvvert r\vvvert_{\mathcal{T}_{n}}\}$

2. (2)

   Let $f\in\mathcal{T}_{n}$ with $\vvvert f\vvvert_{\mathcal{T}_{n}}=1$. Then there exists a $k$-algebra automorphism $\tau$ of $\mathcal{T}_{n}$ such that $\tau(f)$ is regular in $z_{n}$.

([BGR, Theorem 5.2.1.2], [FvdP, Theorem 3.1.1])