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Linear estimate for the number of zeros of abelian integrals of Hamiltonian system with singular line. (Chinese. English summary) Zbl 1389.34103

Summary: It is proved that the number of zeros of abelian integral for the Hamiltonian system \[ \dot x=xy(2+2x-3y),\, \dot y =-y^2(1+2x-y) \] with singular line under perturbations of polynomials with degree \(n\) is not more than \(7\left[\frac{n}{4}\right]+8\) (taking into account the multiplicity).

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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