具有幂零奇点的七次Hamilton系统Abel积分的零点个数估计

摘要

In this paper,we study the number of zeros for Abel integrals of Hamil-ton system of seven degree with nilpotent singularities. By using the Picard-Fuchs equationI method,we derive that the number of zeros of Abel integrals I(h) =∮Γb g(x,y)dx-f(x,y)dy on the open interval (0,1/4) is at most 3[(n-1)/4],where Γh is an oval lying on the alge-braic curve H(x,y) = x2+y2-x2 = h,h ∈ (0,1/4),f(x,y) =∑(1≤4i+4j+1≤n x4i+1y4j and g(x,y)=∑(1≤4i+4j+1≤n x4iy4j+1 are polynomials of x and y of degrees not exceeding n.%本文研究了具有幂零奇点的七次Hamilton系统的Abel积分的零点个数问题.利I用Picard-Fuchs方程法,得到了Abel积分I(h)=∮Γh g(x,y)dx?f(x,y)dy在(0,1/4)上零点个数B(n)≤3[(n-1)/4],其中Γh是H(x,y)=x4+y4?x8=h,h∈(0,1/4),所定义的卵形线f(x,y)=∑(1≤4i+4j+1≤n)aijx4i+1y4j和g(x,y)=∑(1≤4i+4j+1≤n)bijx4iy4j+1是x和y的次数不超过n的多项式.