On the computation of matrices of traces and radicals of ideals

Itnuit Janovitz-Freireich, Bernard Mourrain, Lajos Rónyai, Ágnes Szántó

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Let f1,...,fs∈K{double-struck}[x1,...,xm] 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}[x1,...,xm]/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 {Mxi|i=1,...,m} of the radical √I. We first propose a method using Macaulay type resultant matrices of f1, ..., fs 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 f1,..., fm 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 languageEnglish
Pages (from-to)102-122
Number of pages21
JournalJournal of Symbolic Computation
Volume47
Issue number1
DOIs
Publication statusPublished - Jan 2012

Keywords

  • Matrix of traces
  • Radical of an ideal

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics

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