1998/06/01 by Joan C. Artés, Branko Grünbaum, Jaume Llibre · 1 citation
Mathematics · Computer Science · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Mathematics #Conjecture #Counterexample #Invariant (physics) #Degree (music) #Combinatorics #Polynomial #Invariant polynomial #Differential (mechanical device) #Discrete mathematics #Matrix polynomial #Mathematical analysis #Physics #Mathematical physics
paper · pdf · doi:10.2140/pjm.1998.184.207
openalex publication_date 1998/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (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: (n) is 2n + 1 if n is even, and (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 (5) = 14. Moreover, we present counterexamples to the conjecture for n 6, 7, . . . , 20. We also show that 2n + 1 (n) 3n -1 if n is even, and that 2n + 2 (n) 3n -1 if n is odd.