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Trace et calcul résiduel : une nouvelle version du théorème d'Abel inverse, formes abéliennes

Abstract : We use residue calculus for an effective computation of the trace of a meromorphic form Φ on an analytic hypersurface V and we obtain an algebraic characterization of trace-forms. We prove by this way a stronger version of the global Abel-inverse theorem of Henkin-Passare : the current [V ] ∧ Φ is algebraic if and only if its Abel-transform A(Φ ∧ [V ]) is a rational form in variables not corresponding to the hillside. The proof uses an algebraic mechanism of inversion and a differential equation of a shock wave type satisfied by trace's coefficients. We show the link of this theorem with Wood's theorem, giving a simple criterion for a family of germs of analytic hypersurface to be interpolated by an algebraic hypersurface. Furthermore, we obtain a new method to calculate the dimension of the vector space of maximal abelian forms on an algebraic projective hypersurface.
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https://hal-normandie-univ.archives-ouvertes.fr/hal-02137298
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Submitted on : Wednesday, May 22, 2019 - 9:05:44 PM
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Martin Weimann. Trace et calcul résiduel : une nouvelle version du théorème d'Abel inverse, formes abéliennes. Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc 2007. ⟨hal-02137298⟩

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