On the computation of matrices of traces and radicals of ideals

Abstract

AbstractLet f1,…,fs∈K[x1,…,xm] be a system of polynomials generating a zero-dimensional ideal I, where K is an arbitrary algebraically closed field. We study the computation of “matrices of traces” for the factor algebra A≔K[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 PKm. 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