On the computation of matrices of traces and radicals of ideals

Let f1,…,fs∈K[x1,…,xm]f1,…,fs∈K[x1,…,xm] be a system of polynomials generating a zero-dimensional ideal II, where KK is an arbitrary algebraically closed field. We study the computation of “matrices of traces” for the factor algebra A≔K[x1,…,xm]/IA≔K[x1,…,xm]/I, i.e. matrices with entries which are trace functions of the roots of II. Such matrices of traces in turn allow us to compute a system of multiplication matrices {Mxi∣i=1,…,m}{Mxi∣i=1,…,m} of the radical I.We first propose a method using Macaulay type resultant matrices of f1,…,fsf1,…,fs and a polynomial JJ to compute moment matrices, and in particular matrices of traces for AA. Here JJ is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when II has finitely many projective roots in PKm. We also extend previous results which work only for the case where AA is Gorenstein to the non-Gorenstein case.The second proposed method uses Bezoutian matrices to compute matrices of traces of AA. Here we need the assumption that s=ms=m and f1,…,fmf1,…,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.

► New method to compute moment and trace matrices from Macaulay and Bezout matrices. ► Upper bounds on the degree needed for the Macaulay matrix. ► Extend the results to the non-Gorenstein case. ► Application to compute the radical or the approximate radical of a zero-dimensional ideal.