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Sur le théorème des fonctions composées différentiables

Jean-Jacques Risler — 1982

Annales de l'institut Fourier

Soit f : X Y un morphisme propre relativement algébrique entre espaces semi-analytiques. On montre que si 𝒞 ( Y ) désigne l’anneau des fonctions de classe 𝒞 sur Y , l’image par f de 𝒞 ( Y ) est fermée dans 𝒞 ( X ) muni de sa topologie naturelle d’espace de Frechet ; ceci généralise un résultat précédent de J.-C. Tougeron (lui-même généralisant un résultat de Glaeser) qui traite du cas semi-algébrique. La méthode est tout à fait analogue et utilise des propriétés algébriques de l’anneau des fonctions Nash-analytiques introduit...

Multiplicity of polynomials on trajectories of polynomial vector fields in C 3

Andrei GabrielovFrédéric JeanJean-Jacques Risler — 1998

Banach Center Publications

Let ξ be a polynomial vector field on n with coefficients of degree d and P be a polynomial of degree p. We are interested in bounding the multiplicity of a zero of a restriction of P to a non-singular trajectory of ξ, when P does not vanish identically on this trajectory. Bounds doubly exponential in terms of n are already known ([9,5,10]). In this paper, we prove that, when n=3, there is a bound of the form p + 2 p ( p + d - 1 ) 2 . In Control Theory, such a bound can be used to give an estimate of the degree of nonholonomy...

Multi-Harnack smoothings of real plane branches

Pedro Daniel González PérezJean-Jacques Risler — 2010

Annales scientifiques de l'École Normale Supérieure

Let Δ 𝐑 2 be an integral convex polygon. G. Mikhalkin introduced the notion of, a class of real algebraic curves, defined by polynomials supported on Δ and contained in the corresponding toric surface. He proved their existence, viamethod, and that the topological type of their real parts is unique (and determined by Δ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch ( C , 0 ) . We introduce the class ofsmoothings of ( C , 0 ) by passing through...

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