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Questions tagged [nonarchimedian-analysis]

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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 ...
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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 ...
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1answer
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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} ...
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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 $$f(X)=\sum_J a_JX^J\in \mathbb{k}[[X_1,\dots,X_n]]$$ ...
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0answers
36 views

Structure of units in unramified extension of $\mathbb{Q}_2$

Let $K \mid \mathbb{Q}_2$ be a finite unramified extension of fields of degree $f$, i.e., $2 \in \mathbb{Z}_2$ is a uniformizer in $\mathcal{O}_K$ and $\mathcal{O}_K/2 \mathcal{O}_K \cong \mathbb{F}_{...
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3answers
113 views

Does the above non-Archimedean but ordered field satisfy Nested interval property?

Consider the ordered non-Archimedean field $ \mathbb{R}(t)$, the field of rational function. My question is: $ \text{Does the above non-Archimedean but ordered field satify Nested interval property?} ...
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1answer
47 views

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}} $ ....
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1answer
36 views

Consider the p-adic field $ \ \mathbb{Q}_p \ $ . Define $ \operatorname{ord}_p(x) \ $ to be the p-adic valuation of $ \ x \ $

Consider the p-adic field $ \ \mathbb{Q}_p \ $ . Define $ \operatorname{ord}_p(x) \ $ to be the $p$-adic valuation of $ \ x \ $ by $ \operatorname{ord}_p(x)=\max \{r : \ p^r \ \text{ divides } \ x \} ...
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1answer
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Example of a Map of Banach Spaces over a Non-Archimedian Field with Non-Closed Image

Over archimedian fields, examples of maps $ f \colon X \to Y $ of Banach spaces with non-closed image are well-known, e.g. the inclusion $ \ell^1 \hookrightarrow \ell^2 $ is such an example (which can ...