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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 commonroots V(I)={xi_1,...,xi_k}.  In this paper we show how to construct and certify the rational entries ofHermite matrices for I from the approximate roots {z_1, ...,z_k}. When I isnon-radical, we give methods to construct and certify Hermite matrices for theradical of I from approximate roots. Furthermore, we use signatures of theseHermite matrices to give rational certificates of non-negativity of a givenpolynomial over a (possibly positive dimensional) real variety, as well ascertificates that there is a real root within an epsilon distance from a givenpoint z in Q^n.

Certified Hermite Matrices from Approximate Roots