一类具有幂零中心四次Hamiltonian的Abelian积分的零点个数

The Number of Zeros of Abelian Integrals for a Kind of Quartic Hamiltonians with a Nilpotent Center

该文证明了Hamiltonian H(x,y)=-x^2+ax^2y^2+bx^4+cy^4的Abelian积分在区间(c/(a^2-4bc),0)上零点的个数不超过3n+3[(n-1)/4]+14(计重数),其中a>0,b<-2,c<0,a^2>4bc.

In this paper,we prove that the number of zeros of Abelian integral for the Hamiltonians H（x,y） =-x2＋ ax2y2＋bx4＋cy4on the interval（0,c/（a2-4bc）） is not more than 3n＋3[n-1/4]＋14（taking into account the multiplicity）,where a0,b-2,c 0,a24bc.