Rational limit cycles on generalized Bernouilli polynomial equations

We determine the maximum number of rational limit cycles of the generalized Bernouilli polynomial equations a(x)dy/dx = A(x)yn + B(x)y, where a(x), A(x), and B(x) are real polynomials with a(x)A(x) ≢ 0, n ≥ 3. In particular, we show that when n = 3, there are equations with six rational limit cycles. We also show that the addressed problem can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then, we approach these equations by applying several tools; in particular, some developed to study extending Fermat problems for polynomial equations.