Certified Hermite matrices from approximate roots

Let I=〈f1,…,fm〉⊂Q[x1,…,xn] be a zero dimensional radical ideal defined by polynomials given with exact rational coefficients. Assume that we are approximations {z1,…,zk}⊂Cn for the common roots {ξ1,…,ξk}=V(I)⊆Cn. In this paper show how to construct and certify entries of Hermite matrices I from approximate {z1,…,zk}. When is non-radical, give methods roots. Furthermore, use signatures these certificates non-negativity polynomial over (possibly positive dimensional) real variety, as well there root within an ε distance point z∈Qn.