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

  • Katsusuke NabeshimaEmail author
  • Shinichi Tajima
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


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.


Gröbner bases Extended ideal membership problems Comprehensive Gröbner systems Parametric syzygy systems 



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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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Socio-Arts and SciencesTokushima UniversityTokushimaJapan
  2. 2.Graduate School of Pure and Applied SciencesUniversity of TsukubaTsukubaJapan

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