Moment matrices, trace matrices and the radical of ideals

Details Moment matrices, trace matrices and the radical of ideals Moment matrices, trace matrices and the radical of ideals

2008-11-29

arxiv.org/abs/0812.0088v1

nternational Conference on Symbolic and Algebraic Computation (2008) 125-132

Computer Science

Symbolic Computation

Scientific paper

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 $\mathbb{P}^m_\CC$.

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