Polynomial ODES with many real ovals in the same complex solution

Alcides Lins Neto

Keywords: polynolmial foliations | real ovals

Let $Fol_{\mathbb{R}}(2,d)$ be the space of real algebraic foliations of degree $d$ in $\mathbb{R} P(2)$. For fixed $d$, let $Int_{\mathbb{R}}(2,d)=\{\mathcal{F}\in Fol_{\mathbb{R}}(2,d)\,|\,\mathcal{F}$ has a non-constant rational first integral$\}$. Given $\fa\in Int__{\mathbb{R}}(2,d)$, with primitive first integral $G$, set $O(\mathcal{F})=$ number of real ovals of the generic level $(G=c)$. Let $O(d)=sup\{O(\mathcal{F})\,|\,\mathcal{F}\in Int_{\mathbb{R}}(2,d)\}$. The main purpose of this paper is to prove that $O(d)=+\infty$ for all $d\ge5$.

Anexos:

• algebrcycles.pdf