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Usage of Modular Techniques for Efficient Computation of Ideal Operations

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Abstract

Modular techniques are widely applied to various algebraic computations. In this paper, we discuss how modular techniques can be efficiently applied to computation of Gröbner basis of the ideal generated by a given set, and extend the techniques for further ideal operations such as ideal quotient, saturation and radical computation. In order to make modular techniques very efficient, we focus on the most important issue, how to efficiently guarantee the correctness of computed results. Unifying notions of luckiness of primes proposed by several authors, we give precise and comprehensive explanation on the techniques which shall be a certain basis for further development.

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Acknowledgements

The authors would like to thank the referees for their helpful comments to improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Rikkyo University, 3-34-1 Nish-Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

    Masayuki Noro & Kazuhiro Yokoyama

Authors
  1. Masayuki Noro
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  2. Kazuhiro Yokoyama
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Correspondence to Kazuhiro Yokoyama.

Additional information

This work was supported by JSPS KAKENHI Grant Number 15K05008.

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Noro, M., Yokoyama, K. Usage of Modular Techniques for Efficient Computation of Ideal Operations. Math.Comput.Sci. 12, 1–32 (2018). https://doi.org/10.1007/s11786-017-0325-1

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  • DOI: https://doi.org/10.1007/s11786-017-0325-1

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