|Title||An interactive user interface for division algorithms and the Buchberger algorithm.|
|Author(s)|| Hiromasa Nakayama|
|Type||Book, Chapter in Book, Conference Proceeding|
|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, alg (, ). 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 (, , ). However, in the ring 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.