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Solving Extended Ideal Membership Problems in Rings of Convergent Power Series via Gröbner Bases

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Mathematical Aspects of Computer and Information Sciences (MACIS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9582))

Abstract

An extended ideal membership algorithm is considered in the ring of convergent power series. It is shown that the problem for zero-dimensional ideals in a local ring can be solved in a polynomial ring. The key of the proposed method is the use of ideal quotients in polynomial rings. A new algorithm is given to solve the extended ideal membership problems in local rings. A generalization of the resulting algorithm to ideals with parameters is also described.

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Notes

  1. 1.

    The degree reverse lex. monomial order with the coordinate (x, y) or (x, y, z), is used in the implementation of ExtIMP.

  2. 2.

    syz \((g_1,g_2,\ldots ,g_r)\) outputs a standard basis of the module of syzygies w.r.t. the generators \(g_1,g_2,\ldots ,g_r\) where \(g_1,g_2,\ldots ,g_r \in \mathbb {Q}[x]\). Thus, the command syz outputs the similar results. For each \(i \in \{1,\ldots , 8\}\), syz \((h,\frac{\partial f_i}{\partial x},\frac{\partial f_i}{\partial y},\frac{\partial f_i}{\partial z})\) (or syz \((h,\frac{\partial f_i}{\partial x},\frac{\partial f_i}{\partial y})\)) has been executed in Table 1.

  3. 3.

    The negative degree reverse lex. monomial order with the coordinate (x, y) or (x, y, z), is used in Singular’s command syz.

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Acknowledgments

We thank referees for careful reading our manuscript and for giving useful comments. This work has been partly supported by JSPS Grant-in-Aid for Young Scientists (B) (No.15K17513) and Grant-in-Aid for Scientific Research (C) (No.15K04891).

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Authors and Affiliations

  1. Institute of Socio-Arts and Sciences, Tokushima University, 1-1 Minamijosanjima, Tokushima, Japan

    Katsusuke Nabeshima

  2. Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Japan

    Shinichi Tajima

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

Editor information

Editors and Affiliations

  1. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Hamburg University of Technology, Hamburg, Germany

    Siegfried M. Rump

  3. New York University, New York, New York, USA

    Chee K. Yap

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Nabeshima, K., Tajima, S. (2016). Solving Extended Ideal Membership Problems in Rings of Convergent Power Series via Gröbner Bases. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_22

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  • DOI: https://doi.org/10.1007/978-3-319-32859-1_22

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