Certified Hermite matrices from approximate roots

Creative Commons License

AYYILDIZ AKOĞLU T., Szanto A.

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

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 117
  • Basım Tarihi: 2023
  • Doi Numarası: 10.1016/j.jsc.2022.12.001
  • Dergi Adı: Journal of Symbolic Computation
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH, DIALNET
  • Sayfa Sayıları: ss.101-118
  • Anahtar Kelimeler: Symbolic-numeric computation, Polynomial systems, Approximate roots, Hermite matrices, Certification
  • Karadeniz Teknik Üniversitesi Adresli: Evet

Özet

© 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.