Abstract
We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center-focus problem asks for conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both of these questions for the case in which the Hamiltonian is hyperelliptic. As a by-product, we solve the corresponding problems for the 0-dimensional Abelian integrals defined by Gavrilov and Movasati.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Arnold, Arnold’ problems, Translated and revised edition of the 2000 Russian original. With a preface by V. Philippov, A. Yakivchik and M. Peters, Springer-Verlag, Berlin; PHASIS, Moscow, 2004.
P. Bonnet and A. Dimca, “Relative differential forms and complex polynomials,” Bull. Sci. Math., 124:7 (2000), 557–571.
M. Briskin, J.-P. Françoise, and Y. Yomdin, “Generalized moments, center-focus conditions, and compositions of polynomials,” in: Operator theory, system theory and related topics (Beer-Sheva/Rehovot, 1997), Oper. Theory Adv. Appl., vol. 123, Birkhäuser, Basel, 2001, 161–185.
M. Briskin and Y. Yomdin, “Tangential version of Hilbert 16th problem for the Abel equation,” Mosc. Math. J., 5:1 (2005), 23–53.
C. Christopher, “Abel equations: composition conjectures and the model problem,” Bull. London Math. Soc., 32:3 (2000), 332–338.
J. D. Dixon and B. Mortimer, Permutation Groups, Graduate Texts in Math., vol. 163, Springer-Verlag, New York, 1996.
S. Evdokimov and I. Ponomarenko, “A new look at the Burnside-Schur theorem,” Bull. London Math. Soc., 37 (2005), 535–546.
O. Forster, Lectures on Riemann Surfaces, Graduate Texts in Math., vol. 81, Springer-Verlag, New York-Berlin, 1981.
J.-P. Françoise, “The successive derivatives of the period function of a plane vector field,” J. Differential Equations, 146:2 (1998), 320–335.
L. Gavrilov, “Petrov modules and zeros of Abelian integrals,” Bull. Sci. Math., 122:8 (1998), 571–584.
L. Gavrilov and H. Movasati, “The infinitesimal 16th Hilbert problem in dimension zero,” Bull. Sci. Math., 131:3 (2007), 242–257; http://arxiv.org/abs/math.CA/0507061.
M. Hall, Theory of Groups, Macmillan, New York, 1959.
Y. Il′yashenko, “The origin of limit cycles under perturbation of the equation dw/dz = R z /R w , where R(z,w) is a polynomial,” Mat. Sb., 78:3 (1969), 360–373; English transl.: Math. USSR-Sb., 7:3 (1969), 353–364.
M. Muzychuk and F. Pakovich, “Solution of the polynomial moment problem,” Proc. Lond. Math. Soc., 99:3, 633–657; http://arxiv.org/abs/0710.4085v1.
F. Pakovich, “A counterexample to the ‘composition conjecture’,“ Proc. Amer. Math. Soc., 130:12 (2002), 3747–3749.
F. Pakovich, “On polynomials orthogonal to all powers of a Chebyshev polynomial on a segment,” Israel J. Math., 142 (2004), 273–283.
S. Yakovenko, “A geometric proof of the Bautin theorem. (English summary) Concerning the Hilbert 16th problem,” in: Amer. Math. Soc. Transl. Ser. 2, vol. 165, Amer. Math. Soc., Providence, RI, 1995, 203–219.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional′nyi Analiz i Ego Prilozheniya, Vol. 44, No. 1, pp. 27–43, 2010
Original Russian Text Copyright © by C Christopher and P. Mardešić
The authors would like to thank the Universities of Bourgogne and Plymouth, respectively, for their kind hospitality during the preparation of this work.
Rights and permissions
About this article
Cite this article
Christopher, C., Mardešić, P. The monodromy problem and the tangential center problem. Funct Anal Its Appl 44, 22–35 (2010). https://doi.org/10.1007/s10688-010-0003-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-010-0003-4