no. 2
Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields
Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 611-652.

Soit 𝒜 l’espace vectoriel des intégrales abéliennes

I(h)=∫∫{H≤h}R(x,y)dx∧dy,h∈[0,h˜]

où H(x,y)=(x 2 +y 2 )/2+... est un polynôme réel fixé, R(x,y) est un polynôme réel quelconque, et {H≤h} est l’intérieur de l’ovale de H qui contient l’origine et tend vers lui quand h→0. Nous démontrons que si H(x,y) est un polynôme quasi-homogène avec des points critiques de Morse, alors 𝒜 est un ℝ[h]-module libre de type fini, dont nous calculons le rang. Nous trouvons les générateurs de 𝒜 dans le cas où H est de degré trois. Ce résultat est ensuite appliqué à l’étude des perturbations polynomiales de degré n 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 n-1.

Let 𝒜 be the real vector space of Abelian integrals

I(h)=∫∫{H≤h}R(x,y)dx∧dy,h∈[0,h˜]

where H(x,y)=(x 2 +y 2 )/2+... is a fixed real polynomial, R(x,y) is an arbitrary real polynomial and {H≤h}, h∈[0,h ˜], is the interior of the oval of H which surrounds the origin and tends to it as h→0. We prove that if H(x,y) is a semiweighted homogeneous polynomial with only Morse critical points, then 𝒜 is a free finitely generated module over the ring of real polynomials ℝ[h], and compute its rank. We find the generators of 𝒜 in the case when H is an arbitrary cubic polynomial. Finally we apply this in the study of degree n 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 n-1.

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     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/}
}
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Gavrilov, Lubomir. Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields. Annales de l'Institut Fourier, Tome 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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