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 $x^{\mathrm{\prime }}=a_{1}x-y-a_3{x}^2+(2a_2+a_5)xy+a_6{y}^2$, $y’= x+a_1 y + a_2x^2+(2 a_3+a_4)xy -a_2y^2$. In particular, we study the semi-varieties defined in terms of the parameters $a_1,a_2,\dots ,a_6$ where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.