科研之友
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摘要
An explicit upper bound Z(2, n) <= n + m - 1 is derived for the number of zeros of Abelian integrals M(1)(h) = closed integral(gamma(h)) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where gamma(h) is an oval lying on the algebraic curve H(lambda) = (1/2)x(2)+(1/2)y(2)-(1/3)x(3)-lambda y(3) = h, P(x, y), Q(x, y) are polynomials of x and y, and max{deg P(x, y), degQ(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M(1)(h, lambda) near lambda = 0 and assume (partial derivative(m)/partial derivative lambda(m)) M(1)(h, lambda)vertical bar(lambda=0) not vanish identically.
