zbMATH — the first resource for mathematics

Redundant Picard-Fuchs system for abelian integrals. (English) Zbl 1011.37042
The authors find an explicit system of Picard-Fuchs differential equations satisfied by abelian integrals of monomial forms and give bounds for its coefficients. These results allow the authors to prove the existence of upper bounds for the number of zeros of abelian integrals on a positive distance from the critical locus. As a consequence they give a partial solution to the tangential Hilbert 16th problem. The second part of the paper is devoted to discuss an equivariant formulation of the problem.

37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14H70 Relationships between algebraic curves and integrable systems
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
Full Text: DOI arXiv
[1] Arnold, V.I.; Guseın-Zade, S.M.; Varchenko, A.N., Singularities of differentiable maps. vol. II, monodromy and asymptotics of integrals, (1988), Birkhäuser Boston · Zbl 0659.58002
[2] Balser, W., Formal power series and linear systems of meromorphic ordinary differential equations, (2000), Springer-Verlag New York · Zbl 0942.34004
[3] Bonnesen, T.; Fenchel, W., Theorie der konvexen Körper, (1971), Chelsea Bronx · Zbl 0906.52001
[4] Brieskorn, E., Die monodromie der isolierten singularitäten von hyperfläschen, Manuscripta math., 2, 103-161, (1970) · Zbl 0186.26101
[5] Chademan, A., Bounded algebraic critical values, J. sci. univ. tehran int. ed., 1, 43-50, (1996)
[6] Gavrilov, L., Petrov modules and zeros of abelian integrals, Bull. sci. math., 122, 571-584, (1998) · Zbl 0964.32022
[7] Garvrilov, L., Abelian integrals related to Morse polynomials and perturbations of plane Hamiltonian vector fields, Ann. inst. Fourier, Grenoble, 49, 611-652, (1999) · Zbl 0924.58077
[8] Girard, F.; Jebrane, M., Majorations affines du nombre de zéros d’intégrales abéliennes pour LES hamiltoniens quartiques elliptiques, Ann. fac. sci. Toulouse math. (6), 7, 671-685, (1998) · Zbl 1080.37587
[9] Givental, A.B., Sturm’s theorem for hyperelliptic integrals, Algebra anal., 1, 95-102, (1989) · Zbl 0724.58026
[10] Grünbaum, B., Borsuk’s problem and related questions, Proc. sympos. pure math., (1963), Amer. Math. Soc. Providence, p. 271-284 · Zbl 0151.29101
[11] Horozov, E.; Iliev, I.D., Linear estimate for the number of zeros of abelian integrals with cubic Hamiltonians, Nonlinearity, 11, 1521-1537, (1998) · Zbl 0921.58044
[12] Il’yashenko, Yu.; Yakovenko, S., Counting real zeros of analytic functions satisfying linear ordinary differential equations, J. differential equations, 126, 87-105, (1996) · Zbl 0847.34010
[13] Khovanskii, A., Real analytic manifolds with the property of finiteness, and complex abelian integrals, Funkt. anal. prilozhen., 18, 40-50, (1984)
[14] Khovanskii, A., Fewnomials, Translations of mathematical monographs, 88, (1991), American Mathematical Society Providence
[15] Novikov, D.; Yakovenko, S., Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems, Electron. res. announc. amer. math. soc., 5, 55-65, (1999) · Zbl 0922.58076
[16] Novikov, D.; Yakovenko, S., Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. inst. Fourier, 49, 563-609, (1999) · Zbl 0947.37008
[17] Novikov, D.; Yakovenko, S., Simple exponential estimate for the number of real zeros of complete abelian integrals, Ann. inst. Fourier (Grenoble), 45, 897-927, (1995) · Zbl 0832.58028
[18] Novikov, D.; Yakovenko, S., Meandering of trajectories of polynomial vector fields in the affine n-space, Publ. mat., 41, 223-242, (1997) · Zbl 0878.34026
[19] Dung Tráng, Lê; Ramanujam, C.P., The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. math., 98, 67-78, (1976) · Zbl 0351.32009
[20] Levin, B.Ya., Distribution of zeros of entire functions, Transl. of math. monographs, 5, (1964), Amer. Math. Soc. Providence · Zbl 0152.06703
[21] Pham, F., Singularités des systèmes différentiels de gauss – manin, Progress in mathematics, (1979), Birkhäuser Boston · Zbl 0524.32015
[22] Petrov, G., Complex zeros of an elliptic integral, Funkt. anal. prilozhen., 21, 87-88, (1987) · Zbl 0625.33001
[23] Roitman, M.; Yakovenko, S., On the number of zeros of analytic functions in a neighborhood of a Fuchsian singular point with real spectrum, Math. res. lett., 3, 359-371, (1996) · Zbl 0871.34005
[24] Sebastiani, M., Preuve d’un conjecture de Brieskorn, Manuscripta math., 2, 301-308, (1970) · Zbl 0194.11402
[25] Varchenko, A., Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles, Anal. prilozhen, 18, 14-25, (1984) · Zbl 0545.58038
[26] Yakovenko, S., On functions and curves defined by ordinary differential eqiations, ()
[27] Zariski, O., Studies in equisingularity, III. saturation of local rings and equisingularity, Amer. J. math., 90, 961-1023, (1968) · Zbl 0189.21405
[28] Yulin, Zhao; Zhifen, Zhang, Linear estimate of the number of zeros of abelian integrals for a kind of quartic Hamiltonians, J. differential equations, 155, 73-88, (1999) · Zbl 0961.34016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.