Approximate radical of ideals with clusters of roots

Itnuit Janovitz-Freireich, L. 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 International Symposium on Symbolic and Algebraic Computation, ISSAC
Pages146-153
Number of pages8
Volume2006
Publication statusPublished - 2006
EventInternational Symposium on Symbolic and Algebraic Computation, ISSAC 2006 - Genova, Italy
Duration: Jul 9 2006Jul 12 2006

Other

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

Fingerprint

Factorization
Polynomials

Keywords

  • Algorithms
  • Theory

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

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

Approximate radical of ideals with clusters of roots. / Janovitz-Freireich, Itnuit; Rónyai, L.; Szántó, Ágnes.

Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC. Vol. 2006 2006. p. 146-153.

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

Janovitz-Freireich, I, Rónyai, L & Szántó, Á 2006, Approximate radical of ideals with clusters of roots. in Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC. vol. 2006, pp. 146-153, International Symposium on Symbolic and Algebraic Computation, ISSAC 2006, Genova, Italy, 7/9/06.
Janovitz-Freireich I, Rónyai L, Szántó Á. Approximate radical of ideals with clusters of roots. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC. Vol. 2006. 2006. p. 146-153
Janovitz-Freireich, Itnuit ; Rónyai, L. ; Szántó, Ágnes. / Approximate radical of ideals with clusters of roots. Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC. Vol. 2006 2006. pp. 146-153
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