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Polynomial differential equations with many real ovals in the same algebraic complex solution. (English) Zbl 1273.32039

Summary: Let \(\operatorname{Fol}_{\mathbb{R}}(2,d)\) be the space of real algebraic foliations of degree \(d\) in \(\mathbb{R} \mathbb{P}(2)\). For fixed \(d\), let \[ \operatorname{Int}_{\mathbb{R}}(2,d)=\big\{\mathcal{F}\in \operatorname{Fol}_{\mathbb{R}}(2,d)\mid \mathcal{F} \text{ has a non-constant rational first integral}\big\}. \]
Given \(\mathcal{F}\in \operatorname{Int}_\mathbb{R}(2,d)\) with primitive first integral \(G\), set \(O(\mathcal{F})=\) number of real ovals of the generic level \((G=c)\). Let \(O(d)=\sup\{ O(\mathcal{F})\mid \mathcal{F}\in \operatorname{Int}_{\mathbb{R}}(2,d)\}\). The main purpose of this paper is to prove that \(O(d)=+\infty\) for all \(d\geq5\).

MSC:

32S65 Singularities of holomorphic vector fields and foliations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
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References:

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