# Questions tagged [nonarchimedian-analysis]

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9 questions
0answers
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### absolute value, sup(v(x+1)) bounded if $\mathbb{N}\cdot K$ is bounded

Let $K$ be a field and $v: K \mapsto \mathbb{R}_+$ a map such that $v(x)=0 \Leftrightarrow x=0$ and $v(x \cdot y)=v(x) \cdot v(y)$. Assume that $v$ is bounded on $\mathbb{N}\cdot 1 \subset K$. Is it ...
0answers
34 views

### Reference Request: Manifold theory when $\mathbb{R}$ is replaced by a complete ordered field

Manifolds are typically defined as follows: A topological space $M$ is called a manifold, if for all $x\in M$, there exists an open set $U \ni x$ such that $U$ is homeomorphic to $\mathbb{R}^n$, for ...
1answer
30 views

### An equality of polynomials

Let $\mathcal{O}$ be the ring of valuation integers for a field complete with respect to a non-arch valuation | |. $f(X) \in \mathcal{O}[x]$. Let $f_j(X)$ be defined by the identity \begin{equation} ...
2answers
51 views

### $p$-adic power series and its maximum in the unit ball

Let $\mathbb{K}$ be an algebraically closed field with a complete absolute value and denote by $R$ its valuation ring. Consider a power series $$f(X)=\sum_J a_JX^J\in \mathbb{k}[[X_1,\dots,X_n]]$$ ...
0answers
36 views

1answer
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### How to write disc of convergence in $(1)$ in the following form $|x|_p < K$

p-adic field or Non-Archimedian valued field $\mathbb{Q}_p$: If a power series of the form $\ \sum_{n \geq 0} a_n (x-3)^n$ in $\mathbb{Q}_p$ has radius of convergence $\ R=p^{-\frac{2}{p-1}}$ ....
1answer
36 views