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Enhancements to Lazard’s Method for Cylindrical Algebraic Decomposition

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Computer Algebra in Scientific Computing (CASC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12291))

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Abstract

In 1994 Daniel Lazard proposed an improved method for constructing a cylindrical algebraic decomposition (CAD) from a set of polynomials, which recent work has, finally, fully validated. Lazard’s method works for any set of input polynomials, but is less efficient than the method of Brown (2001) which, however, fails for input sets that are not “well-oriented”. The present work improves Lazard’s method so that it is as efficient for well-oriented input as Brown’s method, while retaining its infallibility. Justifying these improvements requires novel and non-trivial mathematics.

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Notes

  1. 1.

    The system \(a c - 1 = 0 \wedge a c + b^2 = 0\) is easily shown to be unsatisfiable by a number of means, including simply substituting for linearly occurring variables.

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Author information

Authors and Affiliations

  1. United States Naval Academy, Annapolis, U.S.A.

    Christopher W. Brown

  2. Macquarie University, Sydney, Australia

    Scott McCallum

Authors
  1. Christopher W. Brown
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  2. Scott McCallum
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Corresponding author

Correspondence to Christopher W. Brown .

Editor information

Editors and Affiliations

  1. University of Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Brown, C.W., McCallum, S. (2020). Enhancements to Lazard’s Method for Cylindrical Algebraic Decomposition. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_8

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  • DOI: https://doi.org/10.1007/978-3-030-60026-6_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-60025-9

  • Online ISBN: 978-3-030-60026-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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