论文信息 - Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians

Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians

An explicit upper boundZ.3;n/6 5nC15 is derived for the number of the zeros of the integral h!I.h/D R HDhg.x;y/dx f.x;y/dy of degreen polynomialsf;g, on the open interval 6 for which the cubic curvefHDhg contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals RR H<hx i y j dx dy ,i;jD 0; 1; generating the module of complete Abelian integrals I.h/ (over the ring of polynomials in h).

Emil Horozovyx | Iliya D Ilievzk

[1] Uuliî Il'yashenko,et al. Double exponential estimate for the number of zeros of complete Abelian integrals , 1995 .

[2] G. S. Petrov,et al. Number of zeros of complete elliptic integrals , 1984 .

[3] Lubomir Gavrilov,et al. Petrov modules and zeros of Abelian integrals , 1998 .

[4] Iliya D. Iliev,et al. On the Number of Limit Cycles in Perturbations of Quadratic Hamiltonian Systems , 1994 .

[5] Iliya D. Iliev,et al. Perturbations of quadratic centers , 1998 .

[6] Lubomir Gavrilov. Nonoscillation of Elliptic Integrals Related to Cubic Polynomials with Symmetry of Order Three , 1998 .

[7] Sergei Yakovenko,et al. Complete Abelian integrals as rational envelopes , 1994 .

[8] Alexander Varchenko,et al. Estimate of the number of zeros of an abelian integral depending on a parameter and limit cycles , 1984 .

[9] Joe W. Harris,et al. Principles of Algebraic Geometry , 1978 .

[10] Jean-Pierre Francoise,et al. Successive derivatives of a first return map, application to the study of quadratic vector fields , 1996, Ergodic Theory and Dynamical Systems.