Tangential Center Problem for a Family of Non-generic Hamiltonians

The tangential center problem was solved by Yu. S. Ilyashenko in the generic case Mat Sbornik (New Series), 78, 120, 3,360-373, (1969). With the aim of having well-understood models of non-generic Hamiltonians, we consider here a family of non-generic Hamiltonians, whose Hamiltonian is of the form , where f (j) are real polynomials of degree 1. For this family, the genericity assumption of transversality at infinity fails and the coincidence of the critical values for different critical points is allowed. We consider some geometric conditions on these polynomials in order to compute the orbit under monodromy of their vanishing cycles. Under those conditions, we provide a solution of the tangential center problem for this family.