1. Certified Hermite Matrices from Approximate Roots
- Author
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Tulay Ayyildiz Akoglu and Agnes Szanto
- Subjects
Computational Mathematics ,Mathematics - Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Computer Science::Symbolic Computation ,Algebraic Geometry (math.AG) - Abstract
Let I= 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.
- Published
- 2021