About a conjecture on quadratic vector fields

Abstract

The Lotka–Volterra system of autonomous differential equations consists in three homogeneous polynomial equations of degree 2 in three variables. This system, or the corresponding vector field LV ( A , B , C ), depends on three non-zero (complex) parameters and may be written as LV ( A , B , C )= V x ∂ x + V y ∂ y + V z ∂ z where V x =x(Cy+z), V y =y(Az+x), V z =z(Bx+y). As LV ( A , B , C ) is homogeneous, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space C 3 , or curves in the corresponding projective plane CP (2) , such that the trajectories of the vector field are completely contained in a leaf. An homogeneous first integral of degree 0 is then a non-constant function on the set of all leaves of the previous foliation. Trying to classify all values of the triple ( A , B , C ) for which LV ( A , B , C ) has an homogeneous Liouvillian first integral of degree 0, we discovered the interesting family of Lotka–Volterra systems: SLV l =LV 2,B l =− 2l+1 2l−1 ,1/2 , l∈ N ★ . This family provides a negative answer to the conjecture that there exists a uniform bound M 2 such that, if some homogeneous three-variable vector field of degree 2 has a particular solution (an irreducible Darboux polynomial in our words) of degree at least M 2 , then the vector field is rationally integrable (has an homogeneous rational first integral of degree 0). Indeed: • SLV l has no homogeneous rational first integral of degree 0. • SLV l has an irreducible Darboux polynomial f l of degree m =2 l : (∗) x(y/2+z) ∂f l ∂ x +y(2z+x) ∂f l ∂ y +z(B l x+y) ∂f l ∂ z =((l−1)y+2lz)f l .