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The number of zeros of an abelian integral for a class of planar quadratic integrable systems with one centre. (Chinese. English summary) Zbl 1274.34096

Summary: We obtain the finite generators of the abelian integral \[ I(h)=\oint_{\varGamma_h}(M(x, y)g(x, y))\text{d}x-(M(x, y)f(x, y))\text{d}y, \] where \(\varGamma_h\) is a family of closed ovals defined by \(H(x, y)=x^k(\frac 12 y^2+Ax^2+Bx+C)=h,\;h\in \varSigma,\;k\) is a positive integer, \(\varSigma\) is the open interval on which \(\varGamma_h\) is defined, \(f(x, y)\) and \(g(x, y)\) are real polynomials in \(x\) and \(y\) of degrees, not exceeding \(n\). An upper bound of the number of zeros of abelian integral \(I(h)\) for the above system with one centre is given by its algebraic structure for a special case.

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
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