Studying the Upper Bounds of the Numbers of Zeros of Abelian Integrals by the Law of Polynomials

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For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree [Formula: see text], the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.