Certified Hermite Matrices from Approximate Roots

TL;DR

This paper presents methods to construct and certify Hermite matrices from approximate roots of polynomial ideals, enabling rational certificates for polynomial non-negativity and real root proximity.

Contribution

It introduces techniques to build and certify Hermite matrices from approximate roots, including for non-radical ideals, and applies signatures for certificates of non-negativity and root proximity.

Findings

Constructed Hermite matrices from approximate roots.

Certified non-negativity of polynomials over real varieties.

Provided certificates for real roots near given points.

Abstract

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 common roots V(I)={xi_1,...,xi_k}. 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 the radical of I from 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 in Q^n.