Moment matrices, trace matrices and the radical 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. Assume that the factor algebra $\text{A}=\mathbb{K}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_m]\mathrm{/}\text{I}$ is Gorenstein and that we have a bound $\delta >0$ such that a basis for $\text{A}$ can be computed from multiples of $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_s$ of degrees at most $\delta$. We propose a method using Sylvester or Macaulay type resultant matrices of $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_s$ and $J$, where $J$ is a polynomial of degree $\delta$ generalizing the Jacobian, to compute moment matrices, and in particular matrices of traces for $\text{A}$. These 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 $\sqrt{\text{I}}$, following the approach in the previous work by Janovitz-Freireich, R'{o}nyai and Sz'ant'o. Additionally, we give bounds for $\delta$ for the case when $\text{I}$ has finitely many projective roots in ${\mathbb{P}}_{\text{CC}}^m$.