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

Abstract

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 $d\ge5$.