Certified Hermite matrices from approximate roots

Let I = ( f1, ... , fm) C Q[x1, ... , xn] be a zero dimensional radical ideal defined by polynomials given with exact rational coefficients. Assume that we are given approximations {z1, ..., zk} c Cn for the common roots {xi 1, ... , xi k} = V(I) C Cn. In this paper we show how to construct and certify the rational entries of Hermite matrices for I from the approximate roots {z1, ... , zk}. When I is non-radical, we give methods to construct and certify Hermite matrices for ,/I from the 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 E Qn.(c) 2022 Elsevier Ltd. All rights reserved.