一类扰动超椭圆Hamilton系统的Abel积分零点个数上确界

The sharp bound on the number of zeros of Abelian integral for a perturbation of hyper-elliptic Hamiltonian system

本文研究Abel积分Γh(a0+a1x+a2x2+a3x3)ydx零点个数上确界,其中Γh是超椭圆Hamilton量H(x,y)=1/2y2+9/2x2+5x3+7/4x4+1/5x5的闭代数曲线族.根据Abel积分生成元的Chebyshev理论和Abel积分的渐进展开式,结合多项式符号计算技术证明3是Abel积分零点个数的一个上界,并且可以达到3个零点.

In this paper, we study the number of zeros of the Abelian integral Γh（a0＋a1x＋a2x2＋a3x3）ydx defined on the compact level curves Fh of the quintic hyper-elliptic Hamiltonian：H（x,y）=1/2y^2＋9/2x^2＋5x^3＋7/4x^4＋1/5x^5.It is proved that 3 is an upper bound of the number of zeros of the above Abelian integral and 3 zeros can be reached. The proof relies on a Chebyshev criterion （Grau et al. （2011））, some techniques in mathematical mechanization and the asymptotic expansions of Abelian integral.