### 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 International Symposium on Symbolic and Algebraic Computation, ISSAC |

Pages | 146-153 |

Number of pages | 8 |

Volume | 2006 |

Publication status | Published - 2006 |

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

### Other

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

Country | Italy |

City | Genova |

Period | 7/9/06 → 7/12/06 |

### Fingerprint

### Keywords

- Algorithms
- Theory

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - Approximate radical of ideals with clusters of roots

AU - Janovitz-Freireich, Itnuit

AU - Rónyai, L.

AU - Szántó, Ágnes

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Algorithms

KW - Theory

UR - http://www.scopus.com/inward/record.url?scp=33748983348&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748983348&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33748983348

SN - 1595932763

SN - 9781595932761

VL - 2006

SP - 146

EP - 153

BT - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

ER -