In this paper we consider germs of k-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension k and we give a complete modulus of analytic classification under orbital equivalence and a complete modulus of analytic classification under conjugacy. The modulus is an unfolding of the corresponding modulus for the germ of a vector field with a saddle-node. The point of view is to compare the family with a "model family" via an equivalence (conjugacy) over canonical sectors. This is done by studying the asymptotic homology of the leaves and its consequences for solutions of the cohomological equation. This paper is dedicated to the memory of Adrien Douady.

We give unique analytic «normal forms» for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity of saddle-node type having a convergent formal separatrix. We specifically address the problem of computing the normal form explicitly.

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An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some appropriate local chart by a differential equation with algebraic coefficients. We show that there exists at least countably many saddle-node singularities of the complex plane that are not algebraizable.Comment: 11 page

An 0 -type carcinotron (backward -wave oscillator) operating between 850 -1000 GHz was built with the support of the European Space Agency. It generates an RF power level of 0.5 mW minimum across the bandwidth and the maximum RF power reaches 2.3 mW. The operating high voltage varies from 4.9 to 8.8 kV. The total power consumption is less than 200 W. The spectral analysis has been made.The linewidth is less than 1 MHz in free -running operation.

RésuméSoit F un germe de feuilletage singulier du plan complexe. Sous l'hypothèse que F est une courbe généralisée, D. Marín et J.-F. Mattei ont établi l'incompressibilité de F dans un voisinage épointé d'un ensemble fini de courbes analytiques. On montre ici que cette hypothèse ne peut être ignorée, en exhibant divers exemples de feuilletages réduits après un éclatement qui ne satisfont pas cette propriété. Même si nous montrons que les noeuds-cols sont incompressibles individuellement, le fait que leurs feuilles ne se rétractent pas tangentiellement sur toutes les composantes du bord de leur domaine de définition empêche la généralisation totale de la construction de Marín-Mattei. Finalement nous caractérisation une classe presque complète des feuilletages, dits fortement présentables, pour lesquels la construction de la monodromie de Marín-Mattei est possible. AbstractLet F be a germ of a singular foliation of the complex plane. Assuming that F is a generalized curve D. Marín and J.-F. Mattei proved the incompressibility of the foliation in a neighborhood from which a finite set of analytic curves is removed. We show in the present work that this hypothesis cannot be eluded, by building examples of foliations, reduced after one blow-up, for which the property does not hold. Even if we manage to prove that the individual saddle-node foliation is incompressible, their leaves not retracting tangentially on all the components of the definition domain boundary forbids a generalization of Marín-Mattei's construction. We finally characterize a near-complete class of foliations, called strongly presentable, for which the construction of Marín-Mattei's monodromy can be carried out.

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