JOAN C. ARTÉS AND JAUME LLIBRE
Abstract
If $P$ and $Q$ are two real polynomials in the real
variables $x$ and $y$ such that the degree of $P^2+Q^2$ is $2n$,
then we say that the polynomial differential system $x' = P(x,y)$,
$y' = Q(x,y)$ has degree $n$. In the set of all polynomial
differential systems of degree $n>1$ having finitely many invariant
straight lines, let $\alpha(n)$ be the maximum number of invariant
straight lines that they have, and let $\beta(n)$ be the maximum number
of slopes that these invariant straight lines have. Dai Guoren
conjectured that $\beta(n)=2n+(1+(-1)^n)/2$ for $n>2$.
In this paper we prove that the conjecture is true for $n=3,4,5$, and
that it is not true for $n=6,7,\ldots,21$. Moreover, we prove that
$\beta(n)= \alpha(n-1)+1$ and then we refer to \cite {AGL} where
$\alpha(n)$ is studied.}