AN EXPLICIT BOUND FOR THE MULTIPLICITY OF ZEROS OF GENERIC ABELIAN-INTEGRALS

We consider Abelian integrals associated with generic polynomials of a given degree n + 1 and with polynomial 1-forms of degree less-than-or-equal-to n. We give 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 non-conservative polynomial vector field family of degree n. We also give 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.