0000000000092686
AUTHOR
Jean-philippe Rolin
showing 12 related works from this author
Real Algebraic Geometry
2011
144 Pages; Cet ouvrage constitue les actes de la conférence de Géométrie Algébrique Réelle qui a eu lieu à Rennes du 20 au 24 Juin 2011
Multisummability for generalized power series
2022
We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb{R}_{\mathcal{G}}$ and the reduct of $\mathbb{R}_{\mathrm{an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the Gamma function on $(0,\infty)$ and the Zeta function on $(1,\infty)$.
Construction of O-minimal Structures from Quasianalytic Classes
2012
I present the method of constructing o-minimal structures based on local reduction of singularities for quasianalytic classes.
Nature log-analytique du volume des sous-analytiques
2000
Using a preparation theorem for subanalytic functions and Lipschitz stratification for compact subanalytic sets we prove that volumes of slices of globally subanalytic sets and density have a log-analytic nature. We also prove that the set of parameters for which the volume of fiber is finite is globally subanalytic.
Generalised power series solutions of sub-analytic differential equations
2006
Abstract We show that if a solution y ( x ) of a sub-analytic differential equation admits an asymptotic expansion ∑ i = 1 ∞ c i x μ i , μ i ∈ R + , then the exponents μ i belong to a finitely generated semi-group of R + . We deduce a similar result for the components of non-oscillating trajectories of real analytic vector fields in dimension n. To cite this article: M. Matusinski, J.-P. Rolin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).
The Fatou coordinate for parabolic Dulac germs
2017
We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated log scale.
Linearization of complex hyperbolic Dulac germs
2021
We prove that a hyperbolic Dulac germ with complex coefficients in its expansion is linearizable on a standard quadratic domain and that the linearizing coordinate is again a complex Dulac germ. The proof uses results about normal forms of hyperbolic transseries from another work of the authors.
Oscillatory integrals and fractal dimension
2021
Theory of singularities has been closely related with the study of oscillatory integrals. More precisely, the study of critical points is closely related to the study of asymptotic of oscillatory integrals. In our work we investigate the fractal properties of a geometrical representation of oscillatory integrals. We are motivated by a geometrical representation of Fresnel integrals by a spiral called the clothoid, and the idea to produce a classification of singularities using fractal dimension. Fresnel integrals are a well known class of oscillatory integrals. We consider oscillatory integral $$ I(\tau)=\int_{; ; \mathbb{; ; R}; ; ^n}; ; e^{; ; i\tau f(x)}; ; \phi(x) dx, $$ for large value…
Théorème de Gabrielov et fonctions log-exp-algébriques
1997
Resume Nous obtenons le theoreme de Wilkie sur les fonctions log-exp-algebriques du theoreme du complementaire ≪ explicite ≫ de Gabrielov, et de notre presentation geometrique du theoreme de van den Dries, Macintyre et Marker sur les fonctions log-exp-analytiques.
Quasianalytic Denjoy-Carleman classes and o-minimality
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 conditions. Some of these structures do not admit analytic cell decomposition, and they show that there is no largest o-minimal expansion of the real field.
Normal forms and embeddings for power-log transseries
2016
First return maps in the neighborhood of hyperbolic polycycles have their asymptotic expansion as Dulac series, which are series with power-logarithm monomials. We extend the class of Dulac series to an algebra of power-logarithm transseries. Inside this new algebra, we provide formal normal forms of power-log transseries and a formal embedding theorem. The questions of classifications and of embeddings of germs into flows of vector fields are common problems in dynamical systems. Aside from that, our motivation for this work comes from fractal analysis of orbits of first return maps around hyperbolic polycycles. This is a joint work with Pavao Mardešić, Jean-Philippe Rolin and Vesna Župano…
Une structure o-minimale sans décomposition cellulaire
2008
Resume Nous construisons une extension o-minimale du corps des nombres reels qui n'admet pas la propriete de decomposition cellulaire en classe C ∞ . Pour citer cet article : O. Le Gal, J.-P. Rolin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).