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6
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0answers
87 views

What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?

It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...
8
votes
1answer
831 views

If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
1
vote
0answers
77 views

Representing quasianalytic functions in several variables

For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
1
vote
1answer
136 views

Importance of Denjoy-Carleman classes as a class.

Denjoy-Carleman classes of differentiable functions, say in Roumieu's form: Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of ...
5
votes
1answer
251 views

Is there any o-minimal expansion of the real field with functions of growth higher than exponential?

Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there ...
9
votes
2answers
649 views

Intuition behind o-minimal structures.

This is very much the same post as I posted at math.stackexchange. I am following the definitions of an o-minimal system in "Tame Topology and O-minimal Structures" by Lou van den Dries. It is ...
7
votes
0answers
2k views

Why is definable compact equivalent to bounded and closed for sets with o-minimal structures?

We have $M$ an o-minimal structure. $X \in M^n$ with the induced topology. I'm reading an article which shows that $X \in M^n$ is definable compact is equivalent to $X$ being bounded and closed. ...