Separators of fat points in P^n

In this paper we extend the definition of a separator of a point P in P^n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z = m_1P_1 + ... + m_iP_i + .... + m_sP_s in P^n and Z' = m_1P_1 + ... + (m_i-1)P_i + .... + m_sP_s. We associate to P_i a tuple of positive integers of length v = deg Z - deg Z'. We call this tuple the degree of the minimal separators of P_i of multiplicity m_i, and we denote it by deg_Z(P_i) = (d_1,...,d_v). We show that if one knows deg_Z(P_i) and the Hilbert function of Z, one will also know the Hilbert function of Z'. We also show that the entries of deg_Z(P_i) are related to the shifts in the last syzygy module of I_Z. Both results generalize well known results about reduced sets of points and their separators.