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Enhancing the Extended Hensel Construction by Using Gröbner Bases

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Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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Abstract

Contrary to that the general Hensel construction (GHC) uses univariate initial Hensel factors, the extended Hensel construction (EHC) uses multivariate initial Hensel factors determined by the Newton polygon of the given multivariate polynomial. In the EHC so far, Moses-Yun’s (MY) interpolation functions (see the text) are used for Hensel lifting, but the MY functions often become huge when the degree w.r.t. the main variable is large. In this paper, we propose an algorithm which uses, instead of MY functions, Gröbner bases of two initial factors which are homogeneous w.r.t. the main variable and the total-degree variable for sub-variables. The Hensel factors computed by the EHC are polynomials in the main variable with coefficients of mostly rational functions in sub-variables. We propose a method which converts the rational functions into polynomials by replacing the denominators by system variables. Each of the denominators is determined by the lowest order element of a Gröbner basis. Preliminary experiments show that our new EHC method is much faster than the previous one.

Work supported by Japan Society for Promotion of Science KAKENHI Grant number 15K00005.

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Authors and Affiliations

  1. University of Tsukuba, Tsukuba-shi, Ibaraki, 305-8571, Japan

    Tateaki Sasaki

  2. Japanese Association of Mathematics Certification, Ueno 5-1-1, Tokyo, 110-0005, Japan

    Daiju Inaba

Authors
  1. Tateaki Sasaki
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  2. Daiju Inaba
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Correspondence to Tateaki Sasaki .

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Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Sasaki, T., Inaba, D. (2016). Enhancing the Extended Hensel Construction by Using Gröbner Bases. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_29

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_29

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45640-9

  • Online ISBN: 978-3-319-45641-6

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