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International Workshop on Computer Algebra in Scientific Computing

CASC 2020: Computer Algebra in Scientific Computing pp 245–265Cite as

Intrinsic Complexity for Constructing Zero-Dimensional Gröbner Bases

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

Abstract

In this paper, we give a thorough revision of Lakshman’s paper by fixing some serious flaws in his approach. Furthermore, following this analysis, an intrinsic complexity bound for the construction of zero-dimensional Gröbner bases is given. Our complexity bound is in terms of the degree of the input ideal as well as the degrees of its generators. Finally, as an application of the presented method, we exhibit and analyze a (Monte Carlo) probabilistic algorithm to compute the degree of an equidimensional ideal.

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Notes

  1. 1.

    That is all isolated primes of \({\mathfrak {a}}\) share the same dimension. Note that, in general, an equidimensional is not unmixed. A proper ideal is said to be unmixed if its dimension is equal to the dimension of every associated prime of the ideal.

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Acknowledgments

The research was partially supported by the following Iranian, Argentinian and Spanish Grants:

– IPM Grant No. 98550413 (Amir Hashemi)

– UBACyT 20020170100309BA and PICT-2014-3260 (Joos Heintz)

– Spanish Grant MTM2014-55262-P (Luis M. Pardo)

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

    Amir Hashemi

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), 19395-5746, Tehran, Iran

    Amir Hashemi

  3. Departamento de Computación and ICC, UBA-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428, Buenos Aires, Argentina

    Joos Heintz

  4. Depto. de Matemáticas, Estadística y Computación. Facultad de Ciencias, Universidad de Cantabria, Avda. Los Castros s/n, 39071, Santander, Spain

    Luis M. Pardo

  5. Departamento de Matemática and IMAS, UBA-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428, Buenos Aires, Argentina

    Pablo Solernó

Authors
  1. Amir Hashemi
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  2. Joos Heintz
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  3. Luis M. Pardo
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  4. Pablo Solernó
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Corresponding author

Correspondence to Amir Hashemi .

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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Hashemi, A., Heintz, J., Pardo, L.M., Solernó, P. (2020). Intrinsic Complexity for Constructing Zero-Dimensional Gröbner Bases. 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_14

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  • DOI: https://doi.org/10.1007/978-3-030-60026-6_14

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