If P and Q are two real
polynomials in the real variables x and y such that the degree of P2+ Q2 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.
