Poincare bifurcation of a kind of Hamiltonian system under polynomial perturbation

It was proved that the sharp upper bound of the number of zeros ofAbelian integrals and the number of limit cycles bifurcated fromPoincare bifurcation were $B(n)$. An explicit $B(n)$ is derived forthe number of zeros of Abelian integrals $I(h)=oint_{Gamma(h)} f(x,y),dy-g(x,y),dx$ on the open interval $(0,infty)$,where $Gamma(h)$ is an oval lying on the algebraic curve$H(x,y)=x^{2a}/A+y^{2b}/B=h$, $f(x,y)$,$g(x,y)$ arepolynomials of $x$ and $y$, and $max{deg f(x,y),deg g(x,y)}=n$.Assume $I(h)$ not vanish identically, $c=gcd(a,b)$, $lambda=max{a/c,b/c}$,then $B(n)=frac{1}{2}[frac{n-1}{2}]([frac{n-1}{2}]+3)$ for $nleq2lambda$,$B(n)=lambda[frac{n-1}{2}]-frac{1}{2}(lambda-1)(lambda-2)$ for $ngeq2lambda+1$.