Certified Hermite matrices from approximate roots. (July 2023)

• Record Type: Journal Article
• Authors: Ayyildiz Akoglu, Tulay; Szanto, Agnes
• Is Part Of: Journal of symbolic computation. Volume 117(2023)
• Page Start: 101
• Page End: 118
• Publication Date: 2023-07
• Subjects: Symbolic–numeric computation -- Polynomial systems -- Approximate roots -- Hermite matrices -- Certification; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.12.001
• Languages: English
• ISSNs: 0747-7171

Abstract

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 .