Questions tagged [nonarchimedian-analysis]
Nonarchimedean analysis studies the properties of convergence in spaces that do not satisfy the Archimedean property. Examples of such spaces include the $p$-adic numbers and hyperreal and surreal numbers.
64 questions
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Are analytifications in Berkovich geometry strict k-analytic spaces?
Let $k$ be a non-archimedean complete field. Berkovich defines the analytification of a $k$-schemes $X$ which are locally of finite type via the assignment
$\mathrm{Spec}(A) \mapsto \mathcal{M}\mathrm{...
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Fraction field of a complete valued field is complete?
While reading Heuer’s notes on perfectoid spaces, something wasn’t so clear for me on page 2 and I’d love some help. It is said on this page that since the valued ring $R$ is $v_t$ complete, then $K=\...
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Product of power series with integer powers in non-archimedean fields
Suppose we have $K$ a field complete with respect to a non-archimedean absolute value $|\ |$. Consider the power series of the form
$$\sum_{n\in \mathbb{Z}} a_n x^n$$ where $|a_n| \to 0$ if $|n|\to \...
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Whether Cauchy complete and Cantor complete are equivalent in terms of ordered field
For reference, "Cantor complete" means that every nested sequence of bounded closed intervals has non-empty intersection.
It is easy to show that the conditions "Cauchy complete" ...
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What is the quotient norm on the affinoid algebra of Laurent polynomials
Let $E_n = M(T_n)$ be the Berkovich unit $n$-disk.
I am interested in the rational subdomain
$$E_n(1/x_i) = \{ x\in E_n \ \mid \ \lvert 1/x_i \rvert \leq 1 , \ i = 1, \dots, n\}.$$
This is the ...
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Is the intersection of lattices over local ring a lattice?
Let $F$ be a valued field with $O$ the valuation ring (and hence local) of an$\mathbb R$-valued non-archimeadean valuation. We cannot assume Let $M$ be a lattice over $F^2$, that is, a finitely ...
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Robert $p$-adic analysis: Roots of power series are algebraic
Let $K$ be a complete extension of $\Bbb Q_p$ and $f\in K[[t]]$ have radius of convergence $r>0$. In Robert's $p$-adic analysis p. 312 it is claimed that if $f(a)=0$ for some $a\in \Bbb C_p$ with $|...
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Ultrametricity of a pseudometric space when the uniformity induced by the metric is nonarchimedean
I'm trying to understand if/how nonarchimedean uniformities generalize valued fields and ultrametric spaces. It is clear that a nonarchimedean pseudometric induces a nonarchimedean uniformity. Under ...
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Conjecture for Torricelli's parallelogram paradox
Evangelista Torricelli demonstrated a geometrical paradox concerning his view on the homogeneous/heterogeneous debate of the 1600s
(see p.125 Jullien, Vincent, Jullien, Vincent (ed.), Seventeenth-...
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Example of complete zero-characteristic normed field $F$ with open balls not homeomorphic to $F$?
For $F$ a normed field (i.e. a normed ring (Definition 2.1 at the link) that is also a field), let us say $F$ has property "$P$" if any open ball in $F$ (with any center and of any positive ...
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If a normed vector space over a normed field $F$ has "arbitrarily small elements", does $F$ have the same property?
Let $V$ be a nonzero normed vector space over a normed field $F$, using the definitions in the links. (Edit: note the definition used for a "normed field" is just a "normed ring" ...
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Is the $p$-adic unit sphere compact? [duplicate]
Let $p$ be a (nonzero) prime integer, and let $\mathbb{Q}_p$ be the standard field of $p$-adic numbers with the standard $p$-adic norm $||\cdot||$.
Let $S := \{ x \in \mathbb{Q}_p : ||x||=1\}$ be the &...
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What is the right notion of transcendence degree for the fraction field of an affinoid algebra
Let $A$ be a finitely generated $k$-algebra with fraction field $K$.
Then the Krull dimension of $A$ is equal to the transcendence degree of $K$ over $k$.
I would be very interested in any related ...
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Norm $\text{N}_{L/ \Bbb Q_p}(\pi_L)$ of Uniformizer of finite extension of $p$-adics
Let $ K/ \Bbb Q_p$ a finite extension of $p$-adic field $\Bbb Q_p$ of degree $[K:\Bbb Q_p]=n$ and let $ \pi_L$ be a uniformizer of $L$.
Question: Can we say something interesting / "distinguished&...
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What's the importance of the Approximation Theorem - Artin-Whaples Approximation Theorem
The above pictures present the statement of the approximation theorem by Artin-Whaples and its corollary in their paper. I do understand that the theorem implies that we can use one element to ...
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