Infinitesimal and tangential 16-th Hilbert problem on zero-cycles

In this paper, given two polynomials f and g of one variable and a 0-cycle C of f, we consider the deformation f+ϵg. We define two functions: the displacement function Δ(t,ϵ) and its first order approximation: the abelian integral M₁(t). The infinitesimal and tangential 16-th Hilbert problem for zero-cycles are problems of counting isolated regular zeros of Δ(t,ϵ), for ϵ small, or of M₁(t), respectively. We show that the two problems are not equivalent and find optimal bounds, in function of the degrees of f and g, for the infinitesimal and tangential 16-th Hilbert problem on zero-cycles. These two problems are the zero-dimensional analog of the classical infinitesimal and tangential 16-th Hilbert problems for vector fields in the plane.