Countable set of limit cycles for the equation \(dw/dz=P_n(z,w)/Q_n(z,w)\) (Q1972781)

The authors deal with the polynomial differential equations in \(\mathbb{C} P^2\) of the form \[ \frac{dw}{dz}= \frac{P_n(z,w)}{Q_n(z,w)}, \tag{1} \] where \(P_n\), \(Q_n\) are polynomials with complex coefficients of degree at most \(n\). It is known that in a generic case equation (1) possesses a countable set of homologically indpendent limit cycles. The authors prove that the exceptional set -- the set of equations such that they do not possess this property -- has the real codimension two in the space of equations (1).