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Moment matrices, trace matrices and the radical of ideals

Inuit Janovitz-Freireich 1 Agnes Szanto 1 Bernard Mourrain 2 Lajos Ronyai 3
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis (... - 2019), CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : Let $f_1,\ldots,f_s \in \mathbb{K}[x_1,\ldots,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. Assume that the factor algebra $\A=\mathbb{K}[x_1,\ldots,x_m]/\I$ is Gorenstein and that we have a bound $\delta>0$ such that a basis for $\A$ can be computed from multiples of $f_1,\ldots,f_s$ of degrees at most $\delta$. We propose a method using Sylvester or Macaulay type resultant matrices of $f_1,\ldots,f_s$ 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 $\{M_{x_i}|i=1,\ldots,m\}$ of the radical $\sqrt{\I}$, following the approach in the previous work by Janovitz-Freireich, R\'{o}nyai and Szántó. Additionally, we give bounds for $\delta$ for the case when $\I$ has finitely many projective roots in $\mathbb{P}^m_\CC$. \end{abstract}
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Submitted on : Saturday, November 29, 2008 - 1:28:39 PM
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  • HAL Id : inria-00343126, version 1
  • ARXIV : 0812.0088



Inuit Janovitz-Freireich, Agnes Szanto, Bernard Mourrain, Lajos Ronyai. Moment matrices, trace matrices and the radical of ideals. ISSAC, Jul 2008, Linz, Austria. pp.125-132. ⟨inria-00343126⟩



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