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Polynomial differential equations with many real ovals in the same algebraic complex solution

A. Lins Neto

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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 $\operatorname{Int}_{\mathbb{R}}(2,d)=\lbrace\mathcal{F}\in \operatorname{Fol}_{\mathbb{R}}(2,d)\mid \mathcal{F}$ has a non-constant rational first integral$\rbrace$. 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 $O(d)=\sup\lbrace O(\mathcal{F})\mid \mathcal{F}\in \operatorname{Int}_{\mathbb{R}}(2,d)\rbrace$. The main purpose of this paper is to prove that $O(d)=+\infty$ for all $d\ge5$.

Article information

Publ. Mat., Volume 55, Number 2 (2011), 379-399.

First available in Project Euclid: 22 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C09 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.) 37F75: Holomorphic foliations and vector fields [See also 32M25, 32S65, 34Mxx]

Real ovals differential equations


Neto, A. Lins. Polynomial differential equations with many real ovals in the same algebraic complex solution. Publ. Mat. 55 (2011), no. 2, 379--399.

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