An explicit formula for period determinant
Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 887-917.

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.

Nous considérons un polynôme générique à deux variables complexes et une base de cycles dans le premier groupe d’homologie d’une courbe de niveau non singulière. Nous prenons une collection arbitraire de 1-formes polynomiales homogènes de degrés appropriés, de sorte que leurs intégrales le long des cycles de la base forment une matrice carrée (de fonctions multivaluées en la valeur du niveau). Nous calculons le déterminant de cette matrice.

DOI: 10.5802/aif.2204
Classification: 14D05, 32S10, 32S20
Keywords: Complex polynomial in two variables, homology of nonsingular level curve, monodromy, abelian integral, gradient ideal, period determinant
Mot clés : Polynôme complexe à deux variables, homologie d’une courbe de niveau non singulière, monodromie, intégrale abélienne, idéal du gradient, déterminant de périodes
Glutsyuk, Alexey A. 1

1 Laboratoire J.-V.Poncelet (UMI 2615 du CNRS et Université Indépendante de Moscou) Permanent address: CNRS UMR 5669 Unité de Mathématiques Pures et Appliquées École Normale Supérieure de Lyon 46 allée d’Italie 69364 Lyon Cedex 07 (France)
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Glutsyuk, Alexey A. An explicit formula for period determinant. Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 887-917. doi : 10.5802/aif.2204. http://archive.numdam.org/articles/10.5802/aif.2204/

[1] Arnold, V. I.; Varchenko, A. N.; Gussein-Zade, S. M. Singularities of differentiable mappings, Nauka Publ., Moscow, 1982 | MR | Zbl

[2] Deligne, P Équations différentielles à points singuliers réguliers, Lect. Notes in Math., Springer-Verlag, Berlin-New York, 1970 | MR | Zbl

[3] Dimca, A.; Saito, M. Algebraic Gauss-Manin systems and Brieskorn modules, American J. Math., Volume 123 (2001), pp. 163-184 | DOI | MR | Zbl

[4] Erdelyi (Editor), A. Highest transcendental functions, Bateman manuscript project, Volume 1, McGraw Hill, 1953 | Zbl

[5] Gavrilov, L. Petrov modules and zeros of Abelian integrals, Bull. Sci. Math., Volume 122 (1998), pp. 571-584 | DOI | MR | Zbl

[6] Glutsyuk, A. A. Upper bounds of topology of complex polynomials in two variables (To appear in Moscow Math. J.)

[7] Glutsyuk, A. A.; Ilyashenko, Yu. S. Restricted version of the infinitesimal Hilbert 16th problem To appear (in Russian) in Doklady Akademii Nauk (Doklady Mathematics)

[8] Ilyashenko, Yu. S. Example of equations dw/dz=P(z,w)/Q(z,w) having infinite number of limit cycles and arbitrary high Petrovsky-Landis genus, Math. Sbornik, Volume 80 (1969), pp. 388-404 | MR

[9] Ilyashenko, Yu. S. Generation of limit cycles under the perturbation of the equation dw/dz=-R z /R w , where R(z,w) is a polynomial, Math. Sbornik, Volume 78 (1969), pp. 360-373 | MR | Zbl

[10] Lang, S. Algebra, Addison-Wesley, 1965 | MR | Zbl

[11] Milnor, J. Singular points of complex hypersurfaces (in Russian), M. Mir, 1971 | MR | Zbl

[12] Novikov, D. Modules of Abelian integrals and Picard-Fuchs systems, Nonlinearity, Volume 15 (2002), pp. 1435-1444 | DOI | MR | Zbl

[13] Pushkar, I. A. A multidimensional generalization of Ilyashenko’s theorem on abelian integrals (in Russian), Funk. Anal. i Prilozhen., Volume 31 (1997), p. 34-44, 95 Transl. in Funct. Anal. Appl. 31 (1997), 100–108 | MR | Zbl

[14] Varchenko, A. N. Critical values and the determinant of periods (in Russian), Uspekhi Mat. Nauk, Volume 268 (1989), pp. 235-236 Transl. Russian Math. Surveys 44 (1989), 209-210 | MR | Zbl

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