On the Computation of Matrices of Traces and Radicals of Ideals
Abstract
Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zerodimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of "matrices of traces" for the factor algebra $\A := \CC[x_1, ..., x_m]/ \I$, 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 $\{M_{x_i}i=1,...,m\}$ of the radical $\sqrt{\I}$. 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 $\mathbb{P}^m_\CC$. We also extend previous results which work only for the case where $\A$ is Gorenstein to the nonGorenstein 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 $\sqrt{\I}$ are given in terms of Bezoutians.
 Publication:

arXiv eprints
 Pub Date:
 January 2009
 DOI:
 10.48550/arXiv.0901.2778
 arXiv:
 arXiv:0901.2778
 Bibcode:
 2009arXiv0901.2778J
 Keywords:

 Computer Science  Symbolic Computation;
 Mathematics  Commutative Algebra
 EPrint:
 Journal of Symbolic Computation 47, 1 (2012) 102122