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

摘要

An upper bound B(n) <= 7n + 5 is derived for the number of zeros of Abelian integrals I(h) = contuor_(#GAMMA#_h)g(x,y) dy - f(x,y) dx on the open interval #SIGMA#, where #GAMMA#_h is an oval lying on the algebraic curve H(x,y) = 1/2y~2 + U(x) = h, deg U(x) = 4, and #SIGMA#is the maximal interval of existence of #GAMMA#_h. f(x,y), g(x,y) are polynomials of x and y and n=max{deg f(x,y), deg g(x,y)}.