Center conditions: Pull-back of differential equations
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- by Yadollah Zare PDF
- Trans. Amer. Math. Soc. 372 (2019), 3167-3189 Request permission
Abstract:
The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull-back differential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati’s method. The main concepts are the Picard-Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial, and the iterated integral.References
- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR 966191, DOI 10.1007/978-1-4612-3940-6
- Philippe Bonnet, Relative cohomology of polynomial mappings, Manuscripta Math. 110 (2003), no. 4, 413–432. MR 1975095, DOI 10.1007/s00229-002-0313-9
- D. Cerveau and A. Lins Neto, Irreducible components of the space of holomorphic foliations of degree two in $\mathbf C\textrm {P}(n)$, $n\geq 3$, Ann. of Math. (2) 143 (1996), no. 3, 577–612. MR 1394970, DOI 10.2307/2118537
- C. J. Christopher and N. G. Lloyd, Polynomial systems: a lower bound for the Hilbert numbers, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1938, 219–224. MR 1349062, DOI 10.1098/rspa.1995.0081
- H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45–188 (French). MR 1504823
- J. P. Francoise, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 87–96. MR 1375128, DOI 10.1017/S0143385700008725
- A. M. Gabrièlov, Intersection matrices for certain singularities, Funkcional. Anal. i Priložen. 7 (1973), no. 3, 18–32 (Russian). MR 0324066
- Lubomir Gavrilov, Higher order Poincaré-Pontryagin functions and iterated path integrals, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 663–682 (English, with English and French summaries). MR 2188587
- Lubomir Gavrilov, Petrov modules and zeros of Abelian integrals, Bull. Sci. Math. 122 (1998), no. 8, 571–584. MR 1668534, DOI 10.1016/S0007-4497(99)80004-9
- Ju. S. Il′jašenko, The appearance of limit cycles under a perturbation of the equation $dw/dz=-R_{z}/R_{w}$, where $R(z,\,w)$ is a polynomial, Mat. Sb. (N.S.) 78 (120) (1969), 360–373 (Russian). MR 0243155
- Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 2008. MR 2363178, DOI 10.1090/gsm/086
- A. Lins Neto and B. Scárdua, Introdução à teoria das folheações algébricas complexas, IMPA, Rio de Janeiro, Abril de 2011.
- Pavao Mardešić, An explicit bound for the multiplicity of zeros of generic Abelian integrals, Nonlinearity 4 (1991), no. 3, 845–852. MR 1124336
- Hossein Movasati, Abelian integrals in holomorphic foliations, Rev. Mat. Iberoamericana 20 (2004), no. 1, 183–204. MR 2076777, DOI 10.4171/RMI/385
- Hossein Movasati, A course in Hodge theory with emphasis on multiple integrals. To be published, 2017, http://w3.impa.br/~hossein/myarticles/hodgetheory.pdf
- Hossein Movasati, Center conditions: rigidity of logarithmic differential equations, J. Differential Equations 197 (2004), no. 1, 197–217. MR 2030154, DOI 10.1016/j.jde.2003.07.002
- Marco Uribe and Hossein Movasati, Limit cycles, Abelian integral and Hilbert’s sixteenth problem, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2017. 31$^\textrm {o}$ Colóquio Brasileiro de Matemática. MR 3752182
- Hossein Movasati and Isao Nakai, Commuting holonomies and rigidity of holomorphic foliations, Bull. Lond. Math. Soc. 40 (2008), no. 3, 473–478. MR 2418803, DOI 10.1112/blms/bdn029
- Raghavan Narasimhan, Compact analytical varieties, Enseign. Math. (2) 14 (1968), 75–98. MR 242197
- Robert Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics, vol. 164, Birkhäuser Verlag, Basel, 1998. MR 1628014, DOI 10.1007/978-3-0348-8798-4
Additional Information
- Yadollah Zare
- Affiliation: Instituto Nacional de Matemática Pura e Aplicada-IMPA, Rio de Janeiro - RJ, 22460-320, Brazil
- Email: yadollah2806@gmail.com, yadollah@impa.br
- Received by editor(s): July 31, 2017
- Received by editor(s) in revised form: April 5, 2018, May 2, 2018, and July 3, 2018
- Published electronically: June 6, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3167-3189
- MSC (2010): Primary 14P05, 32S65, 34C07, 37F75
- DOI: https://doi.org/10.1090/tran/7660
- MathSciNet review: 3988606
Dedicated: Dedicated to Hossein Movasati