### Abstract

Let f_{1},...,f_{s}∈K{double-struck}[x_{1},...,x_{m}] be a system of polynomials generating a zero-dimensional ideal I, where K{double-struck} is an arbitrary algebraically closed field. We study the computation of "matrices of traces" for the factor algebra A:=K{double-struck}[x_{1},...,x_{m}]/I, i.e. matrices with entries which are trace functions of the roots of I. Such matrices of traces in turn allow us to compute a system of multiplication matrices {M_{xi}|i=1,...,m} of the radical √I. We first propose a method using Macaulay type resultant matrices of f_{1}, ..., f_{s} and a polynomial J to compute moment matrices, and in particular matrices of traces for A. Here J is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when I has finitely many projective roots in P{double-struck}_{K{double-struck}}^{m}. We also extend previous results which work only for the case where A is Gorenstein to the non-Gorenstein case.The second proposed method uses Bezoutian matrices to compute matrices of traces of A. Here we need the assumption that s=m and f_{1},..., f_{m} define an affine complete intersection. This second method also works if we have higher-dimensional components at infinity. A new explicit description of the generators of I are given in terms of Bezoutians.

Original language | English |
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Pages (from-to) | 102-122 |

Number of pages | 21 |

Journal | Journal of Symbolic Computation |

Volume | 47 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2012 |

### Keywords

- Matrix of traces
- Radical of an ideal

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics

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## Cite this

*Journal of Symbolic Computation*,

*47*(1), 102-122. https://doi.org/10.1016/j.jsc.2011.08.020