On the computation of matrices of traces and radicals of ideals
Let f(1,) ..., f(s) is an element of K[x(1,) ..., X-m] be a system of polynomials generating a zero-dimensional ideal l, where K is an arbitrary algebraically closed field. We study the computation of "matrices of traces" for the factor algebra A := K[x(1), ..., x(m)]/l, i.e. matrices with entries which are trace functions of the roots of l. Such matrices of traces in turn allow us to compute a system of multiplication matrices {Mx(i)}i = 1, ..., m} of the radical root l. 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-K(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 root l are given in terms of Bezoutians. (C) 2011 Elsevier Ltd. All rights reserved.
matrix of traces radical of an ideal