Certified Hermite matrices from approximate roots. (July 2023) - Ayyildiz Akoglu, Tulay; Szanto, Agnes
Abstract: Let I = 〈 f 1, …, f m 〉 ⊂ Q [ x 1, …, x n ] be a zero dimensional radical ideal defined by polynomials given with exact rational coefficients. Assume that we are given approximations z 1, …, z k ⊂ C n for the common roots ξ 1, …, ξ k = V ( I ) ⊆ C n . 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 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 ∈ Q n .