On the number of zeros of Abelian integrals for a kind of quartic Hamiltonians

An explicit upper bound B(n)B(n) is derived for the number of zeros of Abelian integrals I(h)=∮Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4)(0,1/4), where ΓhΓh is an oval lying on the algebraic curve H(x,y)=x2+y2-x4+ax2y2+y4H(x,y)=x2+y2-x4+ax2y2+y4 with a>-2,f(x,y) and g(x,y)g(x,y) are polynomials in x and y of degrees not exceeding n  . Assume I(h)I(h) not vanish identically, then B(n)≤3n-14+12n-34+23.