An explicit formula for period determinant

Alexey A. Glutsyuk[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)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 4, page 887-917
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Glutsyuk, Alexey A.. "An explicit formula for period determinant." Annales de l’institut Fourier 56.4 (2006): 887-917. <http://eudml.org/doc/10175>.

@article{Glutsyuk2006,
abstract = {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.},
affiliation = {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)},
author = {Glutsyuk, Alexey A.},
journal = {Annales de l’institut Fourier},
keywords = {Complex polynomial in two variables; homology of nonsingular level curve; monodromy; abelian integral; gradient ideal; period determinant; period matrices; vanishing cycles; Abelian integrals},
language = {eng},
number = {4},
pages = {887-917},
publisher = {Association des Annales de l’institut Fourier},
title = {An explicit formula for period determinant},
url = {http://eudml.org/doc/10175},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Glutsyuk, Alexey A.
TI - An explicit formula for period determinant
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 4
SP - 887
EP - 917
AB - 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.
LA - eng
KW - Complex polynomial in two variables; homology of nonsingular level curve; monodromy; abelian integral; gradient ideal; period determinant; period matrices; vanishing cycles; Abelian integrals
UR - http://eudml.org/doc/10175
ER -

References

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  9. Yu. S. Ilyashenko, Generation of limit cycles under the perturbation of the equation d w / d z = - R z / R w , where R ( z , w ) is a polynomial, Math. Sbornik 78 (1969), 360-373 Zbl0194.40102MR243155
  10. S. Lang, Algebra, (1965), Addison-Wesley Zbl0193.34701MR197234
  11. J. Milnor, Singular points of complex hypersurfaces (in Russian), (1971), M. Mir Zbl0224.57014MR356089
  12. D. Novikov, Modules of Abelian integrals and Picard-Fuchs systems, Nonlinearity 15 (2002), 1435-1444 Zbl1073.14513MR1925422
  13. I. A. Pushkar, A multidimensional generalization of Ilyashenko’s theorem on abelian integrals (in Russian), Funk. Anal. i Prilozhen. 31 (1997), 34-44, 95 Zbl0930.37036MR1475322
  14. A. N. Varchenko, Critical values and the determinant of periods (in Russian), Uspekhi Mat. Nauk 268 (1989), 235-236 Zbl0716.32024MR1023108

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