Complete Abelian integrals for polynomials whose generic fiber is biholomorphic to C∗

Let H be a polynomial of degree m+1 on C2 such that its generic fiber is biholomorphic to C∗, and let ω be an arbitrary polynomial 1-form of degree n on C2. We give an upper bound depending only on m and n for the number of isolated zeros of the complete Abelian integral defined by H and ω.