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A footnote to the Poincaré complete reducibility theorem.

Henrik H. Martens (1992)

Publicacions Matemàtiques

Poincaré's work on the reduction of Abelian integrals contains implicitly an algorithm for the expression of a theta function as a sum of products of theta functions of fewer variables in the presence of reduction. The aim of this paper is to give explicit formulations and reasonably complete proofs of Poincaré's results.

An explicit formula for period determinant

Alexey A. Glutsyuk (2006)

Annales de l’institut Fourier

We consider a generic complex polynomial in two variables and a basis in the first homology group of a nonsingular level curve. We take an arbitrary tuple of homogeneous polynomial 1-forms of appropriate degrees so that their integrals over the basic cycles form a square matrix (of multivalued analytic functions of the level value). We give an explicit formula for the determinant of this matrix.

Finiteness results for Teichmüller curves

Martin Möller (2008)

Annales de l’institut Fourier

We show that for each genus there are only finitely many algebraically primitive Teichmüller curves C , such that (i) C lies in the hyperelliptic locus and (ii) C is generated by an abelian differential with two zeros of order g - 1 . We prove moreover that for these Teichmüller curves the trace field of the affine group is not only totally real but cyclotomic.

Gauss-Manin systems, Brieskorn lattices and Frobenius structures (I)

Antoine Douai, Claude Sabbah (2003)

Annales de l’institut Fourier

We associate to any convenient nondegenerate Laurent polynomial f on the complex torus ( * ) n a canonical Frobenius-Saito structure on the base space of its universal unfolding. According to the method of K. Saito (primitive forms) and of M. Saito (good basis of the Gauss-Manin system), the main problem, which is solved in this article, is the analysis of the Gauss-Manin system of f (or its universal unfolding) and of the corresponding Hodge theory.

Hodge-gaussian maps

Elisabetta Colombo, Gian Pietro Pirola, Alfonso Tortora (2001)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Hyperelliptic action integral

Bernhard Elsner (1999)

Annales de l'institut Fourier

Applying the “exact WKB method” (cf. Delabaere-Dillinger-Pham) to the stationary one-dimensional Schrödinger equation with polynomial potential, one is led to a multivalued complex action-integral function. This function is a (hyper)elliptic integral; the sheet structure of its Riemann surface above the plane of its values has interesting properties: the projection of its branch-points is in general a dense subset of the plane, and there is a group of symmetries acting on the surface. The distribution...

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