On the Computation of Matrices of Traces and Radicals of Ideals

International audienceLet $f_1,\ldots,f_s \in \mathbb{K}[x_1,\ldots,x_m]$ be a system of polynomials generating a zero-dimensional 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, \ldots , 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 $\{M_{x_i}|i=1,\ldots,m\}$ of the radical \sqrt{\I}. We first propose a method using Macaulay type resultant matrices of $f_1,\ldots,f_s$ 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 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 $f_1,\ldots,f_m$ 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

On the computation of matrices of traces and radicals of ideals

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