Complete Abelian integrals for polynomials whose generic fiber is biholomorphic to

Let H be a polynomial of degree m + 1 on C 2 such that its generic fiber is biholomorphic to C ∗ , and let ω be an arbitrary polynomial 1-form of degree n on C 2 . 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 .