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Real Root Isolation of Regular Chains

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Computer Mathematics

Abstract

We present an algorithm RealRootIsolate for isolating the real roots of a polynomial system given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain and a numerical approximation given by intervals. Our algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported.

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Notes

  1. 1.

    Recent investigations of A. Akritas seem to prove that Uspensky only had an incomplete knowledge of Vincent’s paper, from [29, pp. 363–368].

  2. 2.

    The notions of an equiprojectable variety and equiprojectable decomposition are discussed in [8].

  3. 3.

    triangular set in the sense that each polynomial introduces exactly one more variable.

  4. 4.

    the sign of an interval not meeting zero is just the sign of any element of it.

  5. 5.

    otherwise, splittings need to be handled each time a function returns a value.

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Authors and Affiliations

  1. LIFL, Université de Lille 1, 59655, Villeneuve D’Ascq Cedex, France

    François Boulier & François Lemaire

  2. ORCCA, University of Western Ontario (UWO), London, ON, Canada

    Changbo Chen & Marc Moreno Maza

  3. Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, China

    Changbo Chen

Authors
  1. François Boulier
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  2. Changbo Chen
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  3. François Lemaire
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  4. Marc Moreno Maza
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Corresponding authors

Correspondence to François Lemaire or Marc Moreno Maza .

Editor information

Editors and Affiliations

  1. Academy of Mathematics & System Science, Beijing, China

    Ruyong Feng

  2. Department of Mathematics and Computer S, University of Antwerp, Antwerp, Antwerpen, Belgium

    Wen-shin Lee

  3. Department of Mathematical Information S, Tokyo University of Science, Tokyo, Tokyo, Japan

    Yosuke Sato

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Boulier, F., Chen, C., Lemaire, F., Moreno Maza, M. (2014). Real Root Isolation of Regular Chains. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_4

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