An Interactive User Interface for Division Algorithms and the Buchberger Algorithm

  • Hiromasa Nakayama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4151)


Our objectives of building the interactive user interface are as follows:

(1) To select reducers in division algorithms and S-pairs in the Buchberger algorithm interactively.

(2) To visualize division algorithms and the Buchberger algorithm and to understand the algorithms intuitively.

(3) To create a user interface of division algorithms and the Buchberger algorithm without using computer algebra system languages.

Objective (1) has a mathematical background.We have studied and implemented division algorithms and the Buchberger algorithm in the ring of differential operators with rational function coefficients whose denominators do not vanish at the origin, \({\mathcal{D}_{alg}}\) ([5], [3]). In the ring of polynomials and the local ring of that, methods of efficiently computing a remainder and a Gröbner basis have been studied in detail ([1], [2], [4]). However, in the ring \({\mathcal{D}_{alg}}\), methods of those have not been studied in detail. As far as we have known, no system has satisfied our objectives. Therefore, we have designed an interactive user interface as a tool to understand and improve these algorithms. This system is a tool for us to study algorithms, however it may be useful for educational purposes.


Local Ring Interactive User Interface Symbolic Computation Computer Algebra System Division Algorithm 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hiromasa Nakayama
    • 1
  1. 1.Graduate School of Science and TechnologyKobe University 

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