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The infinitesimal 16th Hilbert problem in the quadratic case

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Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let \({X_H} = {H_y}\frac{\partial }{{\partial x}} - {H_x}\frac{\partial }{{\partial y}}\) be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X H in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles.

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Oblatum 23-III-2000 & 19-VI-2000¶Published online: 11 October 2000

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Gavrilov, L. The infinitesimal 16th Hilbert problem in the quadratic case. Invent. math. 143, 449–497 (2001).

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