JOAN C. ARTÉS, BRANKO GRUNBAUM 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 $n$. Let $\alpha(n)$ be the maximum number of invariant
straight lines possible in a polynomial differential systems of
degree $n>1$ having finitely many invariant straight lines.
In the 1980's the following conjecture circulated among mathematicians
working in polynomial differential systems. Conjecture: $\alpha(n)$ is $2n+1$
if $n$ is even, and $\alpha(n)$ is $2n+2$ if $n$ is odd. The conjecture was
established for $n=2,3,4$. In this paper we prove that, in general,
the conjecture is not true for $n>4$. Specifically, we prove that
$\alpha(5)= 14$. Moreover, we present counterexamples to the
conjecture for $n\in \{6,7,\ldots,20\}$. We also show that $2n+1\le
\alpha(n)\le 3n-1$ if $n$ is even, and that $2n+2\le \alpha(n)\le
3n-1$ if $n$ is odd.