一类四次Hamilton函数Abel积分零点个数的估计

证明了Abel积分I(h)=∮ΓhQ(x,y)dx-P(x,y)dy的零点个数的最小上界B(2n+2)=B(2n+1)≤3n/2]+12(n-1)/2]+4(p]表示P的整数部分),这里n是代数曲线H(x,Y)=x2士x4+Y4=h的连通闭分支,h∈E(Γh存在的最大开区间),P(x,y),Q(x,Y)是关于x,y 的次数不超过2n+2或2n+1的实多项式.

Estimation of the Number of Zeros of Abelian Integral for a Kind of Quartic Hamiltonians

It is proved that the supremum of the number of zeros of Abelian integral I(h)=\oint_{{\it \Gamma}_h}Q(x,y)dx-P(x,y)dy satisfies B(2n+2)=B(2n+1)\leq3\frac{n}{2}]+12\frac{n-1}{2}]+4 (p] denotes the entire part of p), where {\Gamma}_h is a compact component of H(x,y)=x^2\pm{x^4}+y^4=h, h\in \Sigma(a maximal open interval of {\Gamma}_h),P(x,y),Q(x,y) are real polynomials of x,y with degree not greater than 2n+2 or 2n+1.