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5
votes
1answer
96 views

Is there an exponential map on (Hahn) ordered fields?

If $F$ is an ordered field and $G$ is an ordered abelian group, one can define the Hahn product $F \boxtimes G$ to be the set of formal Laurent series with coefficients in $F$ and exponents in $G$. It ...
4
votes
0answers
114 views

Relations between two definitions of non-archimedean analytic spaces

I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature. Let us fix a non-archimedean complete valuation field $...
4
votes
1answer
87 views

Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
1
vote
2answers
152 views

Is there a notion of pure dimension for Berkovich analytic space?

For affinoid spaces the definition is similar to algebraic geometry, what about general analytic spaces? I can't find a reference about it. If yes then is the analytification of a variety of pure ...
5
votes
1answer
169 views

Rigid analytic geometry in characterstic 0 vs positive characteristic

This question is motivated purely by curiosity. In algebraic geometry there is a major distinction between the world of characteristic $0$ and that of characteristic $p > 0$ with different methods, ...
0
votes
2answers
174 views

Berthelot functor, rigid analytic space

If $X=\operatorname{Spec} A$, where $A$ is a noetherien, complete local ring, with a finite residual field $\mathbb{F}_p$. We can associate to $A$ a rigid analytic space with two different ways, we ...
5
votes
1answer
140 views

Polynomial inequalities in ordered fields

Let $p(x)$ be a polynomial over an ordered field. If $p'(x)\ge 0$ for all $x$ in an interval over that field, is it true that $p(x)$ is increasing over that interval?
6
votes
2answers
409 views

“Lebesgue-measurable” cardinals and real-closed fields

I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?" Hence, it's also worthwhile to ...
2
votes
1answer
111 views

Determining whether $k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$

My REU partner and I are working on a problem involving iterations of quadratic rational maps over an algebraically closed field $K$ that is complete with respect to a non-trivial non-archimedean ...
4
votes
1answer
138 views

Topology of ring of global sections of finite union of affinoid opens in a rigid analytic space

Let $X$ be a rigid analytic space over a non-Archimedean field $k$. If $U_1,\ldots,U_n\subseteq X$ are affinoid opens, then it's usually not clear whether or not the admissible open $U=U_1\cup\cdots\...
7
votes
0answers
172 views

Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have? More specifically: i) What types of essential singularities can occur? ii) Are ...
1
vote
0answers
76 views

Relative nonarchimedean disks and annuli

Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition. Is there a good notion of closed disk of ...
6
votes
4answers
667 views

A question on non-archimedian Fourier transform

Let $M(n)$ be the vector space of $n\times n$ matrices over a local non-archimedian field $K$. Let $\mathcal S$ denote the space of locally constant compactly supported functions on $M(n)$. Similarly, ...
3
votes
1answer
218 views

is every point of a Berkovich space a Shilov point?

Let $k$ be an algebraically closed non-Archimedean valued field with the value group $\mathbb R$, and let $X$ be a variety over $k$. Is it true that for any point $x \in X^{an}$ of the Berkovich ...
2
votes
1answer
113 views

The target of a regular function in Non-archimedean analytic geometry

Let $(k,|\cdot|)$ be an algebraically closed field, complete wrt a (multiplicative) norm as in the framework of the Berkovich's analytic geometry. Given a commutative Banach $k$-algebra $\mathcal{A}\...
2
votes
2answers
222 views

injective implies completion injective?

Background and definitions. Let $k$ denote a field complete with respect to a non-trivial non-archimedean norm. Let $R$ be the integers in $k$, and say $\pi\in R$ with $0<|\pi|<1$ ($\pi$ doesn'...
2
votes
0answers
139 views

Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
3
votes
1answer
246 views

Spherical completions and flatness

Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? ...
15
votes
1answer
722 views

Why do rigid spaces have “not enough points”?

In Brian Conrad's notes here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...
2
votes
0answers
204 views

Asymptotics vs Puiseux series

Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$. More, we define $X= \{x_i\} \lt Y= \{ ...
11
votes
1answer
549 views

bornological vector spaces over a non-archimedean field

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
2
votes
2answers
357 views

Non-archimedean group over the reals

Hi there, I have a totally ordered group $(\mathbb{R};\leq,\oplus,0,-)$ with the reals as base set satisfying monotonicity, i.e. for all $x,y,z$ we have that if $y\leq z$ then $x\oplus y \leq x\oplus ...
1
vote
2answers
282 views

Non-Archimedean non-standard models for R

Let $\langle \mathbb{R}, 0, 1, +, \cdot, <\rangle$ be the standard model for $R$, and let $S$ be a countable model of $R$ (satisfying all true first-order statements in $R$). Is it true that the ...
10
votes
2answers
743 views

Does Rolle's Theorem imply Dedekind completeness?

I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually ...
9
votes
0answers
243 views

Does antidifferentiability of continuous functions imply Dedekind completeness?

Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
8
votes
4answers
1k views

analysis over non-Archimedean ordered fields

Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...