An explicit bound for the multiplicity of zeros of generic Abelian integrals

The author considers Abelian integrals associated with generic polynomials of a given degree n+1 and with polynomial 1-forms of degree <or=n. He gives an explicit bound C(n) for the multiplicity of zeros of the Abelian integrals considered. A consequence is that C(n) is a bound for the cyclicity of regular cycles of a generic polynomial Hamiltonian vector field of degree n deformed within a nonconservative polynomial vector field family of degree n. He also gives explicit bounds C₀(n) and C_l(n) for the cyclicity of centres and homoclinic loops of generic Hamiltonian vector fields within the considered family.