LINEAR ESTIMATE OF THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR A KIND OF QUINTIC HAMILTONIANS

Abstract. We consider the number of zeros of the integral I(h) = Γ h ω of real polynomial form ω of degree not greater than n over a family of vanishing cycles on curves Γ h : y 2 + 3x 2 − x 6 = h, where the integral is considered as a function of the parameter h. We prove that the number of zeros of I(h), for 0 < h < 2, is bounded above by 2[ n−1 2 ] + 1