ON THE NUMBER OF SLOPES OF INVARIANT

STRAIGHT LINES FOR POLYNOMIAL

DIFFERENTIAL SYSTEMS

Joan C. Artés and Jaume Llibre

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Spain.

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. In the set of all polynomial differential systems of degree n > 1 having finitely many invariant straight lines, let α(n) be the maximum number of invariant straight lines that they have, and let β(n) be the maximum number of slopes that these invariant straight lines have. Dai Guoren conjectured that β(n) = 2n + (1 + (−1)^{n})/2 for n > 2. In this paper we prove that the conjecture is true for n = 3, 4, 5, and that it is not true for n = 6, 7, …, 21. Moreover, we prove that β(n) = α(n − 1) + 1 and then we refer to [AGL] where α(n) is studied.