Let be the real vector space of Abelian integrals
where is a fixed real polynomial, is an arbitrary real polynomial and , , is the interior of the oval of which surrounds the origin and tends to it as . We prove that if is a semiweighted homogeneous polynomial with only Morse critical points, then is a free finitely generated module over the ring of real polynomials , and compute its rank. We find the generators of in the case when is an arbitrary cubic polynomial. Finally we apply this in the study of degree polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle points. We prove that, if the Poincaré-Pontryagin function is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is .
Soit l’espace vectoriel des intégrales abéliennes
où est un polynôme réel fixé, est un polynôme réel quelconque, et est l’intérieur de l’ovale de qui contient l’origine et tend vers lui quand . Nous démontrons que si est un polynôme quasi-homogène avec des points critiques de Morse, alors est un -module libre de type fini, dont nous calculons le rang. Nous trouvons les générateurs de dans le cas où est de degré trois. Ce résultat est ensuite appliqué à l’étude des perturbations polynomiales de degré des champs de vecteurs hamiltoniens quadratiques réversibles, avec un centre et un point selle. Nous démontrons que, si la fonction de Poincaré-Pontryagin n’est pas identiquement nulle, alors la borne supérieure exacte du nombre de cycles limites dans tout domaine compact du plan est égale à .
@article{AIF_1999__49_2_611_0, author = {Gavrilov, Lubomir}, title = {Abelian integrals related to {Morse} polynomials and perturbations of plane hamiltonian vector fields}, journal = {Annales de l'Institut Fourier}, pages = {611--652}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {2}, year = {1999}, doi = {10.5802/aif.1684}, zbl = {0924.58077}, mrnumber = {2000c:34081}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1684/} }
TY - JOUR AU - Gavrilov, Lubomir TI - Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields JO - Annales de l'Institut Fourier PY - 1999 SP - 611 EP - 652 VL - 49 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1684/ DO - 10.5802/aif.1684 LA - en ID - AIF_1999__49_2_611_0 ER -
%0 Journal Article %A Gavrilov, Lubomir %T Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields %J Annales de l'Institut Fourier %D 1999 %P 611-652 %V 49 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1684/ %R 10.5802/aif.1684 %G en %F AIF_1999__49_2_611_0
Gavrilov, Lubomir. Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields. Annales de l'Institut Fourier, Volume 49 (1999) no. 2, pp. 611-652. doi : 10.5802/aif.1684. https://aif.centre-mersenne.org/articles/10.5802/aif.1684/
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