Certified Hermite Matrices from Approximate Roots
Algebraic Geometry
2021-10-22 v1
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.
Cite
@article{arxiv.2110.10313,
title = {Certified Hermite Matrices from Approximate Roots},
author = {Tulay Ayyildiz Akoglu and Agnes Szanto},
journal= {arXiv preprint arXiv:2110.10313},
year = {2021}
}