具有幂零奇点的七次Hamilton系统Abel积分的零点个数估计
ON THE NUMBER OF ZEROS FOR ABEL INTEGRALS OF HAMILTON SYSTEM OF SEVEN DEGREE WITH NILPOTENT SINGULARITIES

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≤naijx4i+1y4j和g（x，y）=∑1≤4i+4j+1≤nbijx4iy4j+1是x和y的次数不超过n的多项式.
In this paper, we study the number of zeros for Abel integrals of Hamilton system of seven degree with nilpotent singularities. By using the Picard-Fuchs equation method, we derive that the number of zeros of Abel integrals I(h)=∫Γh 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 algebraic curve H(x,y)=x4+y4-x8=h,h ∈(0,1/4), f(x,y)=∑1≤4i+4j+1≤naijx4i+1y4j and g(x,y)=∑1≤4i+4j+1≤nbijx4iy4j+1 are polynomials of x and y of degrees not exceeding n