This is a preview of subscription content, log in via an institution to check access.
References
[A] Arnold V. I.: Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl.11 (1977) 85–92
[AO] Arnold V.I., Oleinik O.A.: Topology of real algebraic varieties. Moscow University Vestnik (math. series)6 (1976) 7–17 (Russian)
[AVG] Arnold V.I., Varchenko A.N. and Gussein-Zade S.M.: Singularities of Differentiable Mappings II. Monodromy and Asymptotics of Integrals (Monographs in Mathematics vol. 82). Birkhäuser, Boston, 1986
[Aea] Arnold V.I. et. al.: Some unsolved problems in the theory of differential equations and mathematical physics. Russian Math. Surveys44 (1989) 191–202
[Bg] Bogdanov R.I.: Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues. Seminar Petrovski Proceedings (1976) (Russian). [English translation: Selecta Math. Sovietica1 (1981) 389–421]
[G] Givental A.: The sturm theorem for hyperelliptic integrals. Algebra and Analysis5 (1989) 95–102
[GMM] Gomez Mont X., Muciño J.: Persistent cycles for holomorphic foliations having a meromorphic first integral. In: Holomorphic dynamics, Mexico, 1986 (Lecture Notes in Math., vol. 1345), 129–162
[H] Hartman P.: Ordinary Differential Equations. John Wiley, N. Y.-London-Sydney, 1964
[IY] Il'yashenko Yu. S., Yakovenko S.: Counting real zeros of functions satisfying analytic linear differential equations. J. Differ. Eq. (to appear)
[I1] Il'yashenko Yu. S.: Appearance of limit cycles in perturbation of the equation\(\tfrac{{dw}}{{dz}} = - \tfrac{{R_z }}{{R_w }}\) whereR(z, w) is a polynomial. USSR Math. Sbornik,78 (1969) 360–373; An example of equation\(\tfrac{{dw}}{{dz}} = \tfrac{{P_n (z,w)}}{{Q_n (z,w)}}\) having a countable number of limit cycles and an arbitrarily large genre after Petrovskiî-Landis. Ibib.80 (1969) 388–404
[I2] Il'yashenko Yu. S.: Number of zeros of certain Abelian integrals in the real domain. Funct. Anal. Appl.11 (1978) 78–79
[I3] Il'yashenko Yu. S.: Multiplicity of limit cycles occurring by perturbation of a Hamiltonian differential equationw′=P 2/Q 1 in the real and complex domain. Petrovskii Seminar Proceedings3 (1978) 49–60
[In] Ince E. L.: Ordinary Differential Equations. Dover, 1956
[J] Jacobson N.: Lectures in Abstract Algebra, vol. 2., p. 276. Van Nostrand, Princeton NJ, 1953
[K1] Khovanskiî A. G.: Real analytic varieties with the finiteness property and complex Abelian integrals. Funct. Anal. Appl.,18 (1984) 119–127
[K2] Khovanskiî A. G.: Fewnomials. AMS Publ. Providence, 1991
[Le] Levin B. Ya.: Distribution of zeros of entire functions (Translation of Mathematical Monographs, vol. 5). AMS Publ., Providence, 1964
[Lg] Lang S.: Algebra (3rd edition). Addison-Wesley, 1993.
[M1] Mardešić P.: An explicit bound for the multiplicity of zeros of generic Abelian integrals. Nonlinearity4 (1991) 845–852
[M2] Mardešić P.: The number of limit cycles of polynomial deformations of a Hamiltonian vector field. Ergod. Theory and Dynamical Systems10 (1990) 523–529
[NY] Novikov D., Yakovenko S.: Simple exponential estimate for the number of real zeros of complete Abelian integrals. Comptes Rendus Acad. Sci. Paris (1995) (to appear)
[PL] Petrovskiî I. G., Landis E. M.: On the number of limit cycles for the equation\(\tfrac{{dy}}{{dx}} = \tfrac{{P(x,y)}}{{Q(x,y)}}\), whereP andQ are polynomials of degree 2. Matem. Sbornik37 (1955) 209–250
[Pe1] Petrov G.: Number of Zeros of Complete Elliptic Integrals. Funct. Anal. Appl.18 (1984) 73–74
[Pe2] Petrov G.: Elliptic integrals and their nonoscillation. Funct. Anal. Appl.20 (1986) 46–49
[Po] Poole E. G. C.: Introduction to the Theory of Linear Differential equations. Dover, New York, 1960
[Ri] Roussarie R.: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Brasil Mat.17 (1986) 67–101
[R] Rousseau Ch.: Bifurcation methods in polynomial systems In: Bifurcations and periodic orbits of vector fields, D. Schlomiuk (ed.) Kluwer, Dordrecht, 1993
[V] Varchenko A. N.: Estimate of the number of zeros of Abelian integrals depending on parameters and limit cycles. Funct. Anal. Appl.18 (1984) 98–108
[Y1] Yakovenko S.: Complete Abelian Integrals as Rational Envelopes. Nonlinearity7 (1994)
[Y2] Yakovenko S.: On the number of real zeros of a certain class of Abelian integrals arising in the bifurcation theory. In: Methods of qualitative analysis of differential equations, Gor'kiî, 1984 175–185 (Russian) [English translation: Selecta Matematica Sovietica9 (1990) 255–262
[73] Yakovenko S.: A Geometric Proof of the Bautin Theorem. In: Concerning Hilbert 16th Problem (Advances in the Mathematical Sciences, vol. 27), AMS Publ., Providence, 1995 203–219
Author information
Authors and Affiliations
Additional information
Oblatum 8-VI-1994 & 18-IV-1995
The research was partially supported by the International Science Foundation Grant M98000.
Incumbent of the Elaine Blond Career development chair; the research was partially supported by the Minerva foundation.
Rights and permissions
About this article
Cite this article
Il'yashenko, U., Yakovenko, S. Double exponential estimate for the number of zeros of complete Abelian integrals. Invent Math 121, 613–650 (1995). https://doi.org/10.1007/BF01884313
Issue Date:
DOI: https://doi.org/10.1007/BF01884313