Moment matrices, trace matrices and the radical of ideals

TL;DR

This paper introduces a method to compute the radical of zero-dimensional polynomial ideals using moment and trace matrices derived from Sylvester or Macaulay resultants, applicable over any algebraically closed field.

Contribution

It generalizes the computation of radical ideals by utilizing Sylvester or Macaulay matrices and provides bounds for the degree parameter in the projective root case.

Findings

Method computes multiplication matrices of the radical ideal.

Applicable over arbitrary algebraically closed fields.

Provides bounds for degree parameter δ in projective roots case.

Abstract

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. Assume that the factor algebra $\A=\mathbb{K}[x_1,...,x_m]/\I$ is Gorenstein and that we have a bound $\delta>0$ such that a basis for $\A$ can be computed from multiples of $f_1,...,f_s$ of degrees at most $\delta$. We propose a method using Sylvester or Macaulay type resultant matrices of $f_1,...,f_s$ and $J$, where $J$ is a polynomial of degree $\delta$ generalizing the Jacobian, to compute moment matrices, and in particular matrices of traces for $\A$. These matrices of traces in turn allow us to compute a system of multiplication matrices $\{M_{x_i}|i=1,...,m\}$ of the radical $\sqrt{\I}$, 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 $\I$ has finitely many projective roots in $\mathbb{P}^m_\CC$.