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A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals

  • Jean Bernard LasserreEmail author
  • Monique Laurent
  • Philipp Rostalskl
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 149)

Abstract

In this paper we propose a unified methodology for computing the set V K (I) of complex (K = ℂ) or real (K = ℝ) roots of an ideal R[x], assuming Vk (I ) is finite. We show how moment matrices, defined in terms of a given set of generators of the ideall, can be used to (numerically) find not only the real variety V R (I), as shown in the Authors’ previous work, but also the complex variety V c (I), thus leading to a. unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods arc outlined and their stopping criteria are related.

Key words

Polynomial ideal zero-dimensional ideal complex roots real roots numerical linear algebra 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Jean Bernard Lasserre
    • 1
    Email author
  • Monique Laurent
    • 2
  • Philipp Rostalskl
    • 3
  1. 1.LAAS-CNRS and Institute of MathematicsToulouseFrance
  2. 2.Centrum voor Wiskunde en InformaticaAmsterdamNetherlands
  3. 3.Automatic Control LaboratoryETH ZurichZurichSwitzerland

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