Linear Estimate of the Number of Zeros of Abelian Integrals for a Kind of Quartic Hamiltonians

An upper bound B(n) 7n+5 is derived for the number of zeros of Abelian integrals I(h)= Γh g(x, y) dy−f(x, y) dx on the open interval Σ, where Γh is an oval lying on the algebraic curve H(x, y)= y2+U(x)=h, deg U(x)=4, and Σ is the maximal interval of existence of Γh. f(x, y), g(x, y) are polynomials of x and y and n=max{deg f(x, y), deg g(x, y)}.