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Équation homologique et classification analytique des germes de champs de vecteurs holomorphes de type noeud-col

Abstract : I mainly studied the classification of holomorphic vector fields Z, admitting an isolated saddle-node singularity at (0,0), under the action of the local holomorphic changes of coordinates. The problem can be reduced to the study of two homological equations: one for the classification of the underlying foliation, another one for the classification of the flow for a given foliation. Thus the functional invariants of Martinet/Ramis for foliations are coupled with other functional invariants for vector fields. The invariants express the obstructions to the existence of a solution F to an equation Z(F)=G. By integrating the right side along paths tangent to Z, the obstructions are localized in the non-vanishing of some integrals along asymptotic cycles. This geometrical approach differs from the methods used by Merscheryakova/Voronin to obtain, independantly and about the sime time, similar results. For one, this approach gives rise to a "natural" integral representation of Martinet/Ramis' invariants. By estimating these quantities I finally derive many explicit classes of vector fields (and differential equations) mutually non-conjugated, whereas in other cases I can give normal forms.
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https://tel.archives-ouvertes.fr/tel-00005387
Contributor : Loïc Jean Dit Teyssier <>
Submitted on : Friday, March 19, 2004 - 12:59:59 AM
Last modification on : Thursday, January 7, 2021 - 4:12:37 PM
Long-term archiving on: : Friday, April 2, 2010 - 8:16:47 PM

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  • HAL Id : tel-00005387, version 1

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Loïc Jean Dit Teyssier. Équation homologique et classification analytique des germes de champs de vecteurs holomorphes de type noeud-col. Mathématiques [math]. Université Rennes 1, 2003. Français. ⟨tel-00005387⟩

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