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A Modular Algorithm for Computing the Intersection of a One-Dimensional Quasi-Component and a Hypersurface

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Computer Algebra in Scientific Computing (CASC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14139))

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Abstract

Computing triangular decompositions of polynomial systems can be performed incrementally with a procedure named Intersect. This procedure computes the common zeros (encoded as regular chains) of a quasi-component and a hypersurface. As a result, decomposing a polynomial system into regular chains can be achieved by repeated calls to the Intersect procedure. Expression swell in Intersect has long been observed in the literature. When the regular chain input to Intersect is of positive dimension, intermediate expression swell is likely to happen due to spurious factors in the computation of resultants and subresultants.

In this paper, we show how to eliminate this issue. We report on its implementation in the polynomial system solver of the BPAS (Basic Polynomial Algebra Subprogram) library. Our experimental results illustrate the practical benefits. The new solver can process various systems which were previously unsolved by existing implementations of regular chains. Those implementations were either limited by time, memory consumption, or both. The modular method brings orders of magnitude speedup.

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Notes

  1. 1.

    See Sect. 2 for a review of regular chain theory, including definitions of the terms quasi-component, initial, etc.

  2. 2.

    The differential ideal generated by finitely many differential polynomials is generally not finitely generated, when regarded as an algebraic ideal.

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Authors and Affiliations

  1. Department of Computer Science, The University of Western Ontario, London, Canada

    Alexander Brandt, Juan Pablo González Trochez, Marc Moreno Maza & Haoze Yuan

Authors
  1. Alexander Brandt
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  2. Juan Pablo González Trochez
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  3. Marc Moreno Maza
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  4. Haoze Yuan
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Corresponding author

Correspondence to Alexander Brandt .

Editor information

Editors and Affiliations

  1. University of Lille, CRIStAL, Villeneuve d'Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias Kotsireas

  4. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

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

    Evgenii V. Vorozhtsov

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Brandt, A., González Trochez, J.P., Moreno Maza, M., Yuan, H. (2023). A Modular Algorithm for Computing the Intersection of a One-Dimensional Quasi-Component and a Hypersurface. In: Boulier, F., England, M., Kotsireas, I., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2023. Lecture Notes in Computer Science, vol 14139. Springer, Cham. https://doi.org/10.1007/978-3-031-41724-5_4

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  • DOI: https://doi.org/10.1007/978-3-031-41724-5_4

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