  Prony's method, in its various concrete algorithmic realizations, is
concerned with the reconstruction of a sparse exponential sum from integer
samples. In several variables, the reconstruction is based on finding the
variety for a zero dimensional radical ideal. If one replaces the coefficients
in the representation by polynomials, i.e., tries to recover sparse exponential
polynomials, the zeros associated to the ideal have multiplicities attached to
them . The precise relationship between the coefficients in the exponential
polynomial and the multiplicity spaces are pointed out in this paper.
