The smallest upper bound on the number of zeros of Abelian integrals

Abstract

In this paper, we develop a new method to estimate the smallest upper bound on the number of isolated zeros of Abelian integrals, and give an algebraic criterion for the case of Abelian integrals along energy level ovals of potential systems. As applications of our main result, we first obtain a criterion guaranteeing that two Abelian integrals have the Chebyshev property with accuracy one and give an example of hyperelliptic Abelian integrals. Then we get a criterion that any nontrivial linear combination of three Abelian integrals has at most two isolated zeros. Using this criterion we prove that the smallest upper bound is two for the number of isolated zeros of a non-algebraic Abelian integral along the oval (y2+12)1x2+ln⁡|x|−12=h with 0<h<+∞, which corresponds to the perturbation of a quadratic reversible system with two centers. Moreover, we derive all possible configurations of limit cycles from the Poincaré bifurcation of this perturbation. To our knowledge, these problems can not be solved by other known approaches.