### 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 language | English |
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Title of host publication | Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006 |

Pages | 146-153 |

Number of pages | 8 |

Publication status | Published - Sep 29 2006 |

Event | International Symposium on Symbolic and Algebraic Computation, ISSAC 2006 - Genova, Italy Duration: Jul 9 2006 → Jul 12 2006 |

### Publication series

Name | Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC |
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Volume | 2006 |

### Other

Other | International Symposium on Symbolic and Algebraic Computation, ISSAC 2006 |
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Country | Italy |

City | Genova |

Period | 7/9/06 → 7/12/06 |

### Keywords

- Algorithms
- Theory

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*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).