An Interactive User Interface for Division Algorithms and the Buchberger Algorithm

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.