# O-minimal Analytic Separation of Sets in Dimension 2

@article{Fischer2009OminimalAS, title={O-minimal Analytic Separation of Sets in Dimension 2}, author={Andreas Fischer}, journal={Ann. Pure Appl. Log.}, year={2009}, volume={157}, pages={130-138} }

Abstract We study the Hardy field associated with an o-minimal expansion of the real numbers. If the set of analytic germs is dense in the Hardy field, then we can definably analytically separate sets in R 2 , and we can definably analytically approximate definable continuous unary functions. A similar statement holds for definable smooth functions.

#### Topics from this paper

#### One Citation

Extending piecewise polynomial functions in two variables

- Mathematics
- 2013

We study the extensibility of piecewise polynomial functions defined on closed subsets of R2 to all of R2. The compact subsets of R2 on which every piecewise polynomial function is extensible to R2… Expand

#### References

SHOWING 1-10 OF 28 REFERENCES

Smooth approximation of definable continuous functions

- Mathematics
- 2008

Let M be an o-minimal expansion of the real exponential field which possesses smooth cell decomposition. We prove that for every definable open set, the definable indefinitely continuously… Expand

Smooth functions in o-minimal structures

- Mathematics
- 2008

Abstract Fix an o -minimal expansion of the real exponential field that admits smooth cell decomposition. We study the density of definable smooth functions in the definable continuously… Expand

Rings of analytic functions definable in o-minimal structure

- Mathematics
- 2003

Abstract From the ring theoretical viewpoint, especially from the viewpoint of Real Algebra, we consider the ring of analytic functions definable in a given o-minimal expansion of the real field on a… Expand

Transition maps at non-resonant hyperbolic singularities are o-minimal

- Mathematics
- 2006

Abstract We construct a model complete and o-minimal expansion of the field of real numbers such that, for any planar analytic vector field ξ and any isolated, non-resonant hyperbolic singularity p… Expand

THE REAL FIELD WITH CONVERGENT GENERALIZED POWER SERIES

- Mathematics
- 1998

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn 1 is definable. This… Expand

The Pfaffian closure of an o-minimal structure

- Mathematics
- 1997

Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C1 coefficients are definable.

Quasianalytic Denjoy-Carleman classes and o-minimality

- Mathematics
- 2003

We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure… Expand

Expansions of the Real Field with Power Functions

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 1994

It is shown that the (O-minimal) theory of the ordered field of real numbers augmented by all restricted analytic functions and all real power functions admits elimination of quantifiers and has a universal axiomatization. Expand

The Elementary Theory of Restricted Analytic Fields with Exponentiation

- Mathematics
- 1994

numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the… Expand

Infinite differentiability in polynomially bounded o-minimal structures

- Mathematics
- 1995

Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion 9l of the ordered field of real numbers are shown to have some of the nice properties of real analytic… Expand