On the Computation of Matrices of Traces and Radicals of Ideals

Let $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_s\in \mathbb{K}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_m]$ be a system of polynomials generating a zero-dimensional ideal $\text{I}$, where $\mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of “matrices of traces” for the factor algebra $\text{A}:=\text{CC}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_m]\mathrm{/}\text{I}$, i.e. matrices with entries which are trace functions of the roots of $\text{I}$. Such matrices of traces in turn allow us to compute a system of multiplication matrices $\{M_{x_i}\mathrm{\mid }i=1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$ of the radical $\sqrt{\text{I}}$. We first propose a method using Macaulay type resultant matrices of $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_s$ and a polynomial $J$ to compute moment matrices, and in particular matrices of traces for $\text{A}$. Here $J$ is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when $\text{I}$ has finitely many projective roots in ${\mathbb{P}}_{\text{CC}}^m$. We also extend previous results which work only for the case where $\text{A}$ is Gorenstein to the non-Gorenstein case. The second proposed method uses Bezoutian matrices to compute matrices of traces of $\text{A}$. Here we need the assumption that $s=m$ and $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_m$ 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 $\sqrt{\text{I}}$ are given in terms of Bezoutians.