  Let I=<f_1, ..., f_m> be a zero dimensional radical ideal Q[x_1,...,x_n].
Assume that we are given approximations {z_1,...,z_k} in C^n for the common
roots V(I)={xi_1,...,xi_k}.
  In this paper we show how to construct and certify the rational entries of
Hermite matrices for I from the approximate roots {z_1, ...,z_k}. When I is
non-radical, we give methods to construct and certify Hermite matrices for the
radical of I from approximate roots. Furthermore, we use signatures of these
Hermite matrices to give rational certificates of non-negativity of a given
polynomial over a (possibly positive dimensional) real variety, as well as
certificates that there is a real root within an epsilon distance from a given
point z in Q^n.
