Uniqueness of limit cycles for quadratic vector fields

This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as \begin{document}$x' = a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$\end{document} , \begin{document}$y' = x+a_1 y + a_2x^2+(2 a_3+a_4)xy -a_2y^2$\end{document} . In particular, we study the semi-varieties defined in terms of the parameters \begin{document}$a_1, a_2, ..., a_6$\end{document} where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.