THE REAL FIELD WITH CONVERGENT GENERALIZED POWER SERIES

@article{Dries1998THERF,
  title={THE REAL FIELD WITH CONVERGENT GENERALIZED POWER SERIES},
  author={Lou van den Dries and Patrick Speissegger},
  journal={Transactions of the American Mathematical Society},
  year={1998},
  volume={350},
  pages={4377-4421}
}
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 expansion is polynomially bounded. 
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References

SHOWING 1-10 OF 21 REFERENCES
Logarithmic-Exponential Power Series
We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zetaExpand
Expansions of the Real Field with Power Functions
  • Chris Miller
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 1994
TLDR
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
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 theExpand
Tame Topology and O-minimal Structures
  • L. Dries
  • Mathematics, Computer Science
  • 1998
1. Some elementary results 2. Semialgebraic sets 3. Cell decomposition 4. Definable invariants: Dimension and Euler characteristic 5. The Vapnik-Chernovenkis property in o-minimal structures 6.Expand
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
Recall that a subset of R is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form {~ α ∈ R : p(~ α) = 0}, {~ α ∈ R : q(~ α) > 0} where p(~x), q(~x) areExpand
Geometric categories and o-minimal structures
The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, Bierstone and Milman [1]. Regrettably, certain “nice” sets—like { (x, x) : x > 0 } forExpand
Lectures on Riemann Surfaces
Contents: Covering Spaces.- Compact Riemann Surfaces.- Non-compact Riemann Surfaces.- Appendix.- References.- Symbol Index.- Author and Subject Index.
Semianalytic and subanalytic sets
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I. The Tarski-SeidenbergExpand
Semianalytic and subanalytic sets, Inst
  • Hautes Etudes Sci. Publ. Math
  • 1988
Sur les ensembles semi-analytiques avec condition Gevrey au bord MR 94m:32013 16. , Paramétrisations de petits chemins en géométrie analytique réelle, Singularities and Differential Equations
  • Ann. Sc. Ec. Norm. Sup. Polish Acad. Sci. CMP
  • 1994
...
1
2
3
...