# Quasianalytic Ilyashenko Algebras

@article{Speissegger2018QuasianalyticIA, title={Quasianalytic Ilyashenko Algebras}, author={P. Speissegger}, journal={Canadian Journal of Mathematics}, year={2018}, volume={70}, pages={218 - 240} }

Abstract We construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty $ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation and log-composition; in particular, $\mathcal{F}$ is a Hardy field. Moreover, the field $\mathcal{F}\,\circ \,\left( -\text{log} \right)$ of germs at ${{0}^{+}}$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

#### 5 Citations

Ilyashenko algebras based on transserial asymptotic expansions

- Mathematics
- 2018

We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is… Expand

Analytic continuations of log-exp-analytic germs

- Mathematics
- 2017

We describe maximal, in a sense made precise, analytic continuations of germs at infinity of unary functions definable in the o-minimal structure R_an,exp on the Riemann surface of the logarithm. As… Expand

Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3

- Mathematics
- 2018

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a… Expand

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

- Mathematics
- 2017

New striking analogies between H. Hahn’s fields of generalised series with real coefficients, G. H. Hardy’s field of germs of real valued functions, and J. H. Conway’s field No of surreal numbers,… Expand

Asymptotics of parameterized exponential integrals given by Brownian motion on globally subanalytic sets

- Mathematics
- 2017

We consider parameterized exponential integrals coming from the time evolution of the probability distribution of Brownian motion on globally subanalytic sets. We establish definability results and… Expand

#### References

SHOWING 1-10 OF 21 REFERENCES

Action de derivations irreductibles sur les algebres quasi-regulieres d'Hilbert

- Mathematics
- 2009

We study the action of irreducible derivations X on some Hilbert's quasi-regular algebras QRH of germes at 0 of analytic functions on (U,0), where U is a semi-algebraic set: that is, we show that… Expand

The Riemann mapping theorem for semianalytic domains and o-minimality

- Mathematics
- 2009

We consider the Riemann Mapping Theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (i.e. biholomorphic map) from the upper half… Expand

Logarithmic-exponential series

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 2001

Evidence is given for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields, and its basic properties are established, including the existence of compositional inverses. Expand

Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

- Mathematics
- 1996

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) are… Expand

The Dirichlet problem in the plane with semianalytic raw data, quasi analyticity, and o-minimal structure

- Mathematics
- 2007

We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a… Expand

Finiteness Theorems for Limit Cycles

- 2010

It is known from the time of Poincaré that the Theorem 4 implies Theorems 1-3. The Theorem 3 is the direct consequence of the Theorem 4: The singular point is a particular case of a polycycle. It is… Expand

Real and complex analysis

- Mathematics
- 1966

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures… 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