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On the number of zeros for Abel integrals of Hamilton system of seven degree with nilpotent singularities. (Chinese. English summary) Zbl 1399.34099

Summary: In this paper, we study the number of zeros for Abel integrals of Hamilton systems of seven degree with nilpotent singularities. By using the Picard-Fuchs equation method, we derive that the number of zeros of Abel integrals \(I\left ( h \right) = \oint_{{\Gamma_h}} {g\left ( {x, y} \right)dx - f\left ( {x, y} \right)dy} \) on the open interval \(\left ( {0, \frac{1}{4}} \right)\) is at most \(3\left[ {\frac{{n - 1}}{4}} \right]\), where \({\Gamma_h}\) is an oval lying on the algebraic curve
\[ \begin{aligned} &H\left ( {x, y} \right) = {x^4} + {y^4} - {x^8} = h, \quad h \in \left ( {0, \frac{1}{4}} \right),\\ &f(x, y) = \sum\limits_{1 \leq 4i + 4j + 1 \leq n} {{x^{4i + 1}}{y^{4j}}} \text{ and } g\left ( {x, y} \right) = \sum\limits_{1 \leq 4i + 4j + 1 \leq n} {{x^{4i}}{y^{4j + 1}}} \end{aligned} \]
are polynomials of \(x\) and \(y\) of degrees not exceeding \(n\).

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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