Certified Hermite matrices from approximate roots

Abstract

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 given approximations {z1,…,zk}⊂Cn for the common roots {ξ1,…,ξk}=V(I)⊆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 ε distance from a given point z∈Qn.