C. Christopher, P. Mardešic, “The Monodromy Problem and the Tangential Center Problem”, Funktsional. Anal. i Prilozhen., 44:1 (2010), 27–43; Funct. Anal. Appl., 44:1 (2010), 22

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Funktsional'nyi Analiz i ego Prilozheniya, 2010, Volume 44, Issue 1, Pages 27–43
DOI: https://doi.org/10.4213/faa2980 (Mi faa2980)

This article is cited in 10 scientific papers (total in 10 papers)

The Monodromy Problem and the Tangential Center Problem

C. Christopher^a, P. Mardešic^b

^a School of Mathematics and Statistics, University of Plymouth
^b Institut de Mathématiques de Bourgogne, Unité mixte de recherche 5584 du C.N.R.S., Université de Bourgogne

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DOI: https://doi.org/10.4213/faa2980

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.

Keywords: tangential center, Abelian integral, composition, monodromy.

Received: 08.10.2008

English version:
Functional Analysis and Its Applications, 2010, Volume 44, Issue 1, Pages 22–35
DOI: https://doi.org/10.1007/s10688-010-0003-4

Bibliographic databases:

Document Type: Article

UDC: 517.9

Language: Russian

Citation: C. Christopher, P. Mardešic, “The Monodromy Problem and the Tangential Center Problem”, Funktsional. Anal. i Prilozhen., 44:1 (2010), 27–43; Funct. Anal. Appl., 44:1 (2010), 22–35

Citation in format AMSBIB

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https://doi.org/10.4213/faa2980

https://www.mathnet.ru/eng/faa/v44/i1/p27

This publication is cited in the following 10 articles:

Daniel López-Garcia, Fabricio Valencia, “On the monodromy action for 𝑓(𝑥,𝑦)=𝑔(𝑥)+ℎ(𝑦)”, Proc. Amer. Math. Soc., 2025
J.L. Bravo, P. Mardešić, D. Novikov, J. Pontigo-Herrera, “Infinitesimal and tangential 16-th Hilbert problem on zero-cycles”, Bulletin des Sciences Mathématiques, 2025, 103634
A. Álvarez, J. L. Bravo, C. Christopher, P. Mardešić, “Infinitesimal Center Problem on Zero Cycles and the Composition Conjecture”, Funct. Anal. Appl., 55:4 (2021), 257–271
Garcia D.L., “The Monodromy Problem For Hyperelliptic Curves”, Bull. Sci. Math., 170 (2021), 102998
Pontigo-Herrera J., “Tangential Center Problem For a Family of Non-Generic Hamiltonians”, J. Dyn. Control Syst., 23:3 (2017), 597–622
A. Álvarez, J.L. Bravo, C. Christopher, “On the Trigonometric Moment Problem”, Ergod. Theory Dyn. Syst., 34:1 (2014), 1–20
L. Gavrilov, F. Pakovich, “Moments on Riemann surfaces and hyperelliptic Abelian integrals”, Comment. Math. Helv., 89:1 (2014), 125–155
Alvarez A., Bravo J.L., Mardesic P., “Vanishing Abelian Integrals on Zero-Dimensional Cycles”, Proc. London Math. Soc., 107:6 (2013), 1302–1330
A. Álvarez, J. L. Bravo, P. Mardešić, “Inductive solution of the tangential center problem on zero-cycles”, Mosc. Math. J., 13:4 (2013), 555–583
Francoise J.P., Pakovich F., Yomdin Y., Zhao W., “Moment vanishing problem and positivity: Some examples”, Bull. Sci. Math., 135:1 (2011), 10–32

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Functional Analysis and Its Applications

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