Certified Hermite matrices from approximate roots

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 Hermit...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 117; pp. 101 - 118
Main Authors: Ayyildiz Akoglu, Tulay, Szanto, Agnes
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.07.2023
Subjects:
Approximate roots
Certification
Hermite matrices
Polynomial systems
Symbolic–numeric computation
Approximate roots
Hermite matrices
Polynomial systems
Symbolic–numeric computation
Certification
ISSN:0747-7171, 1095-855X
Online Access:Get full text
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Summary: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.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2022.12.001