On the number of invariant straight lines for polynomial differential systems

If P and Q are two real polynomials in the real variables x and y such that the degree of P 2 +Q is 2n, then we say that the polynomial differential system x′ = P (x, y), y′ = Q(x, y) has degree n. Let α(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: α(n) is 2n+ 1 if n is even, and α(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 α(5) = 14. Moreover, we present counterexamples to the conjecture for n ∈ {6, 7, . . . , 20}. We also show that 2n + 1 ≤ α(n) ≤ 3n − 1 if n is even, and that 2n+ 2 ≤ α(n) ≤ 3n− 1 if n is odd.