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Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians

An explicit upper bound is derived for the number of the zeros of the integral of degree n polynomials f, g, on the open interval for which the cubic curve contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals , i,j=0,1, generating the module of complete Abelian integrals I(h) (over the ring of polynomials in h).