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…
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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…
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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…
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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…
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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…
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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…
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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…
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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 $k$, $U\subseteq X$…
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Over a henselian field, are any two norms on a given finite dimensional vector space equivalent?

Let $k$ be a field with a non-archimedean absolute value. If $k$ is complete, then two norms on a finite dimensional $k$-vector space are always equivalent. This fact is for example commonly invoked in the proof that a complete non-archimedean field…
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Does infinitesimals exists in $p$-adics?

Infinitesimals can't exist in $\mathbb{R},$ since it satisfy the Archimedean Property. That is, given any positive real number $\varepsilon \gt 0$ and any positive real number $M\gt 0,$ there exists a natural number $n$ such that $n\varepsilon \gt…
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Compact subgroups of a p-adic field

Definition: A p-adic field is a finite extension of $Q_p$. Question: Let $E$ be a p-adic field, $G$ is a nontrivial additive compact subgroup of $E$, how to prove: $G$ is isomorphic to $Z_p^n$ for some positive integer $n$. This isomorphism is not…
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Why does $f(x_n)\rightarrow 0$ imply that a subsequence converges to zero of $f$?

Let $f$ be a separable and irreducible polynomial of degree $d\geq 1$ with coefficients in a local field of characteristic $p$, say $K= \mathbb F_p((T))$ and $f\in K[X]$. Assume there is a sequence $(y_n)_{n\in \mathbb N}$ in the separable closure…
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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?} $ Answer: The field of rational function…
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In an affinoid/Tate algebra, can one always find a system of topological generators that have specific properties at prescribed points?

Let $k$ be a complete non-archimedean field with absolute value $| \cdot |$. The Tate algebra over $k$ in the variables $X_1,\ldots,X_n$ is $T_n = k\langle{X_1,\ldots,X_n} \rangle = \{ \sum_{J \geq 0} a_JX^{J}: a_J \in k, |a_J| \to 0 \ \textrm{as} …
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Levi-Civita field vs Puiseux series: why is Cauchy completeness important?

The Levi-Civita field's main claim to fame is that it's the smallest real-closed field which is a proper extension of the reals and which is Cauchy-complete. That last bit is important, or else the Puiseux series would instead take the crown: it is…
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