讨论了首次积分为$H(x,y)=x^k(\frac{1}{2}{y}^2+A{x}^2+Bx+C)$的Abel积分的代数构造,并研究了$k=2$时具有一个中心的平面二次可积系统在$n$次扰动下的Abel积分零点个数上界问题,得到了较小的上界估计.

In this paper, we obtain the finite generators of Abelian integral I(h) =H??h(M(x, y)g(x, y)) dx − (M(x, y)f(x, y)) dy, where ??h is a family of closed ovals defined by
H(x, y) = xk( 12y2+Ax2+Bx+C) = h, h ∈ , k is a positive integer,  is the open interval on which ??h is defined, f(x, y) and g(x, y) are real polynomials in x and y of degrees, not exceeding n. An upper bound of the number of zeros of Abelian integral I(h), for the above system with one centre, is given by its algebraic structure for a special case.In this paper, we obtain the finite generators of Abelian integral I(h) =H??h(M(x, y)g(x, y)) dx − (M(x, y)f(x, y)) dy, where ??h is a family of closed ovals defined by H(x, y) = xk( 12y2+Ax2+Bx+C) = h, h ∈ , k is a positive integer,  is the open interval on which ??h is defined, f(x, y) and g(x, y) are real polynomials in x and y of degrees, not exceeding n. An upper bound of the number of zeros of Abelian integral I(h), for the above system with one centre, is given by its algebraic structure for a special case.