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Solving Parametric Ideal Membership Problems and Computing Integral Numbers in a Ring of Convergent Power Series Via Comprehensive Gröbner Systems

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Abstract

New algorithms are presented for solving ideal membership problems with parameters and for computing integral numbers in a ring of convergent power series. It is shown that the ideal membership problems for zero-dimensional ideals in the ring of convergent power series, can be solved in the polynomial ring. The key idea of the algorithms is computing comprehensive Gröbner systems of ideal quotients in a polynomial ring. Furthermore, new methods for computing integral numbers, in the local ring, are introduced as an application.

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Authors and Affiliations

  1. Graduate School of Technology, Industrial and Social Sciences, Tokushima University, 2-1, Minamijosanjima, Tokushima, Japan

    Katsusuke Nabeshima

  2. Graduate School of Science and Technology, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, Japan

    Shinichi Tajima

Authors
  1. Katsusuke Nabeshima
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  2. Shinichi Tajima
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Correspondence to Katsusuke Nabeshima.

Additional information

This work has been partly supported by JSPS Grant-in-Aid for Scientific Research (Nos. 15KT0102, 15K04891) and JSPS Grant-in-Aid for Young Scientist (B) (No. 15K17513).

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Nabeshima, K., Tajima, S. Solving Parametric Ideal Membership Problems and Computing Integral Numbers in a Ring of Convergent Power Series Via Comprehensive Gröbner Systems. Math.Comput.Sci. 13, 185–194 (2019). https://doi.org/10.1007/s11786-018-0354-4

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  • DOI: https://doi.org/10.1007/s11786-018-0354-4

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