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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.

In $\mathbb{R}$ with the usual absolute value $|\cdot|$, the Archimedean property is the statement that if $0<x<y$, there is an $n\in\mathbb{N}^+$ such that $nx >y$; in a general ordered field, this is replaced with the condition $\underbrace{x+\cdots +x}_{n}>y$. A non-Archimedean space is a space that does not satisfy this property. The most familiar example is the $p$-adic numbers under the valuation metric.

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Can $\pi$ be defined in a p-adic context?

I am not at all an expert in p-adic analysis, but I was wondering if there is any sensible (or even generally accepted) way to define the number $\pi$ in $\mathbb Q_p$ or $\mathbb C_p$. I think that circles, therefore also angles, are problematic in…
doetoe
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Prerequisites for Condensed Mathematics and Analytic Geometry

I'm a student in Algebraic Geometry. I've read chapter 2 and 3 of Hartshorne. I want to study the theory of Condensed Mathematics and Analytic Geometry by Scholze and Clausen. What are the basic prerequisites for understanding the theory? How much…
Luiz Felipe Garcia
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Can inequivalent Krull valuations induce the same topology?

I was working with (Krull) valuations and I have a question I could not answer. Let me recall you some notions first. Let $K$ be a field. Given a totally ordered abelian group $\Gamma$, we say $v: K^\times \to \Gamma$ is a valuation on $K$ if it is…
Don
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How can a field be Cauchy complete and non Archimedean

The Wikipedia page for the completeness of the Real numbers, says that “ there are non Archimedean fields that are ordered and Cauchy complete.” However, in many other places, I’ve read that non Archimedean fields must be incomplete with the example…
Lave Cave
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Can every nonarchimedean ordered field be embedded in some hyperreal field?

Let $F$ be a nonarchimedean ordered field. Is there always a hyperreal field $^*\mathbb{R}$ such that there is an embedding of $F$ in $^*\mathbb{R}$? As far as I understand it, the answer here implies that the answer is positive in the special case…
David M
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Banach-Alaoglu Theorem over spherically complete non-Archimedean fields

About a year ago I asked here whether the Banach-Alaoglu Theorem works over the $p$-adics. The satisfactory answer I got is that the "usual" proof only uses local compactness, and so the Banach-Alaoglu Theorem holds for any local field. Now I would…
frafour
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Is evaluation of formal power series compatible with composition over nonarchimedean complete fields?

In algebraic number theory, one may want to consider a $p$-adic local field and consider the $p$-adic logarithm and $p$-adic exponential function on it. These form inverse homomorphism between a sufficiently higher unit group (multiplicative) and a…
Thorgott
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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" and "Cantor complete" are equivalent in Archimedean ordered fields.…
dy Deng
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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 result where $A$ is an affinoid algebra. I suppose…
Kemal
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Analog of Mahler's theorem over other non archimedean fields

For any continuous function $f:\mathbb{Z}_p \to \mathbb{Q}_p$, Mahler's theorem provides us with a relatively explicit series of polynomials converging uniformly to $f$. Is there any analogue for other non archimedean fields? In particular, what…
ReLonzo
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Why is the generalized definition of valued field automatically non-Archimedean?

The classical definition of a field $K$ with an absolute value $|\cdot|:K\to \mathbb{R}_{\geq 0}$ is that $\forall x,y\in K$ $x=0\Leftrightarrow|x|=0$ $|xy|=|x|\cdot|y|$ $|x+y|\leq |x|+|y|$ If the last one can be replaced by a stronger…
Z Wu
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Analytic functions on spaces over non-Archimedean fields and troubles with totally disconnectedness

I read in several intro scripts on Berkovish spaces that these arose as new approach to analytic geometry over non-archimedean fields. As the main problem in non-archimedean analytic geometry is recognized the observation that analytic…
user267839
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Is $|x+y|_p=\mathrm{max}(|x|_p,|y|_p)$ for $p$-adic norm? Sanity test.

I apologize in advance, if this is a stupid question. I thought about this for some time and don't see any mistake in my reasoning, even after carefully typing up the proof here. I think this is true Claim. For any non-zero $x,y\in\mathbb{Q}$ we…
Tyrell
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Theorems of the type $D\lim_n f_n = \lim_n Df_n$ for non-Archimedean fields

I'm kind of interested in seeing if there is a base-field agnostic setting for calculus, but I don't have much experience with the non-Archimedian case. Let $X,Y$ be Banach spaces over some non-discrete locally compact field $k$, $U\subseteq X$…
s.harp
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Converse to Hensel's Lemma

The following was an exercise in some MIT course notes on p-adic numbers and Hensel's lemma. Let $f\in \Bbb Z_p[X]$. Suppose that $b$ is a simple root of $f$. Prove that for any $a\in \Bbb Z_p$: if $|a-b|_p<|f'(b)|_p$ then $|f(a)|_p<|f'(a)|_p^2$.…
Dr. Heinz Doofenshmirtz
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