# An Interactive User Interface for Division Algorithms and the Buchberger Algorithm

## Abstract

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.

## Keywords

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

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## References

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