ON THE NUMBER OF INVARIANT STRAIGHT LINES

FOR POLYNOMIAL DIFFERENTIAL SYSTEMS

Joan C. Artés^{1}, Branko Grünbaum^{2} and Jaume Llibre^{1}

1 Departament de Matemàtiques, Universitat Autònoma de Barcelona,

Bellaterra 08193, Barcelona, Spain.

2 Department of Mathematics, University of Washington, Seattle, WA 98195–4350, USA.

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 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.