Moment matrices, border bases and real radical computation

Abstract : In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finte. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of \cite{Mourrain2005} are efficient and numerically stable for computing complex roots, algorithms based on moment matrices \cite{LLR07} allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Gröbner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.
Type de document :
Communication dans un congrès
MEGA2011, May 2011, Stockholm, Sweden. MEGA 2011, MEGA2011, pp.1-23
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https://hal.archives-ouvertes.fr/hal-01282654
Contributeur : Lip6 Publications <>
Soumis le : vendredi 4 mars 2016 - 10:08:36
Dernière modification le : mercredi 30 janvier 2019 - 15:30:03

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  • HAL Id : hal-01282654, version 1

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Bernard Mourrain, Monique Laurent, Jean-Bernard Lasserre, Philippe Trébuchet. Moment matrices, border bases and real radical computation. MEGA2011, May 2011, Stockholm, Sweden. MEGA 2011, MEGA2011, pp.1-23. 〈hal-01282654〉

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