Resolution of the Poincaré problem and nonexistence of algebraic limit cycles in family (I) of Chinese classification

Any quadratic system with limit cycles can be written in one of the three families stated by the Chinese classification. In this paper we consider family (I), i.e., x˙=δx-y+ℓx2+mxy+ny2,y˙=x. We show that the degree of its real irreducible invariant algebraic curves is bounded by 3. By the way, we prove that there is not any algebraic limit cycle for this family.