NEW UPPER BOUNDS ON THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR THREE PERTURBED QUADRATIC REVERSIBLE SYSTEMS

For the quadratic reversible systems $r12$, $r16$ and $r18$, when they are perturbed by arbitrary polynomial of degree $n$, we explore the new upper bounds on the number of zeros for their Abelian integrals using the methods of Riccati equation and Picard-Fuchs equation. We consider both the highest degree and the lowest degree of the related polynomials, and more importantly, we notice their law between the terms, and take into account the range of values for variable. Some laws of related polynomials of the systems have been found, therefore, we obtain some new upper bounds, which are better than those obtained before.