On the invariant straight lines of polynomial differential systems (Q1387148)

There is an interesting but difficult problem: to find the maximum number \(\alpha (n)\) of invariant straight lines in the set of all polynomial differential systems of degree \(n>1\) having finitely many invariant straight lines. Many authors proved independently that \(\alpha (2)=5\), \(\alpha (3)=8\), \(\alpha (4)=9\). Later, in 1995, H. Zoladek proved that \(\alpha (n)\leq 3n-1\). In this paper the authors give chiefly the following results: 1. \(2n+1\leq \alpha (n)\leq 3n-1\) for \(n\) even, 2. \(2n+2 \leq\alpha (n)\leq 3n-1\) for \(n\) odd, 3. \(\alpha (5)=14\). It is interesting to note that for \(n=2,3\) and 5, the authors have \(\alpha (n)=3n-1\) but for \(n=4\), \(\alpha (4)=9 <3\times 4-1\). Therefore, one may ask: How this problem relates to number theory?