Approximate radical of ideals with clusters of roots

Itnuit Janovitz-Freireich, Lajos Rónyai, Ágnes Szántó

Research output: Chapter in Book/Report/Conference proceedingConference contribution

9 Citations (Scopus)

Abstract

We present a method based on Dickson's lemma to compute the "approximate radical" of a zero dimensional ideal I in ℂ[x 1 , . . . , x m] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is "global" in the sense that it does not require any local approximation of the zero clusters: it reduces the problem to the computation of the numerical nullspace of the so called "matrix of traces", a matrix computable from the generating polynomials of Ĩ. To compute the numerical nullspace of the matrix of traces we propose to use Gauss elimination with pivoting, and we prove that if Ĩ has k distinct zero clusters each of radius at most s in the ∞-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ε 2. We also prove that the computed approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to 2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.

Original languageEnglish
Title of host publicationProceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006
Pages146-153
Number of pages8
Publication statusPublished - Sep 29 2006
EventInternational Symposium on Symbolic and Algebraic Computation, ISSAC 2006 - Genova, Italy
Duration: Jul 9 2006Jul 12 2006

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
Volume2006

Other

OtherInternational Symposium on Symbolic and Algebraic Computation, ISSAC 2006
CountryItaly
CityGenova
Period7/9/067/12/06

Keywords

  • Algorithms
  • Theory

ASJC Scopus subject areas

  • Mathematics(all)

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    Janovitz-Freireich, I., Rónyai, L., & Szántó, Á. (2006). Approximate radical of ideals with clusters of roots. In Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006 (pp. 146-153). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC; Vol. 2006).