Certified Hermite matrices from approximate roots

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AYYILDIZ AKOĞLU T., Szanto A.

Journal of Symbolic Computation, vol.117, pp.101-118, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 117
  • Publication Date: 2023
  • Doi Number: 10.1016/j.jsc.2022.12.001
  • Journal Name: Journal of Symbolic Computation
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.101-118
  • Keywords: Symbolic-numeric computation, Polynomial systems, Approximate roots, Hermite matrices, Certification
  • Karadeniz Technical University Affiliated: Yes

Abstract

© 2022 Elsevier LtdLet 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.