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Masayuki Noro

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Development of Computer Algebra Software

Masayuki Noro
Degree: Doctor of Engineering (Ehime University)
Research Interests: Computational Algebra
Unit:Number & Equation

Report

The main purpose of our research and development topic, computer algebra software, is to execute symbolic and exact computations in polynomial rings. Because such computations deal with polynomials expressed in a tree structure and arbitrary precision numbers, they require a large amount of main memory and high-speed CPU, which previously forced us to use commercial systems such as Maple and Mathematica on expensive computers. However, rapid improvements in both speed and capacity of personal computers have made it possible to execute large-scale algebraic computations on a personal computer. At the same time, a number of free computer algebra software has been developed around the world and distributed. I was involved in the development of computer algebra software called Risa/Asir at Fujitsu Laboratories Ltd. Initially we were distributing only its executable files free of charge, but when I joined the faculty of Kobe University in 2000, I was granted permission to distribute the software, including its source codes, free of charge.

Figure:Risa/Asir: Computation of b-function

As its name suggests, computer algebra addresses mainly algebraic computation. Therefore, it is naturally well-suited for mathematics, especially algebra. For example, the Groebner basis, which gives a “good” set of generators for polynomial ideals, is finding a wide range of applications for obtaining solutions for polynomial systems and computations of various invariants of algebraic varieties as well as algebraic geometry and algebraic analysis. Nowadays, numerous computer algebra software systems have implemented these functions, which make it easier to conduct a computational experiment. However, often there are cases where a complicated problem is inputted, and the computation does not terminate.

My own interest is to improve algorithms and implementation for these algebraic computations so that wide ranging input can be efficiently computed. In particular, I have been making an effort to improve the efficiency of the Groebner basis computation for more than the last ten years. The Buchberger algorithm for computing the Groebner basis is a simple one, yet it requires a number of improvements before it can be implemented for practical purposes. We have poroposed several new improvements, implemented them, and verified their effectiveness. In my opinion, applications of computation on finite fieds, in particular, are essential for improving the efficiency. I am also interested in the Groebner basis computation in Weyl algebra (ring of differential operators with polynomial coefficients) and its applications. Toshinori Oaku (Tokyo Women’s Christian University) and Nobuki Takayama (Kobe University) have proposed numerous algorithms for D-modules. b-Function plays an important role in these algorithms. We have successfully expanded their targets for computation by improving the b-function computation algorithm. Our success was partly due to my experience with commutative polynomial rings, which I have shared with my group, and our research is an application of the Groebner basis.

Algebraic computation is often considered to have applications only in mathematics; we do not need to look for immediate industrial applications, in my opinion. Unexpected industrial applications similar to number theory in cryptography are always possible. For instance, there are already applications of the Groebner basis to integer programming, statistics, and biology. Takayuki Hibi (Osaka University) is developing new applications from a mathematician’s point of view, and I am looking forward to fruitful results.

Numerous free mathematical software systems, not only computer algebra software, are now available. I have been working with Tatsuyoshi Hamada (Fukuoka University) to assemble many of these free software systems and to distribute them as KNOPPIX/Math, which is a Linux Live DVD. This DVD contains numerous programs useful for applying mathematics, including numerical computation, statistical computation, and visualization. My goal is to offer people not just in mathematics, but also those in other fields, the opportunity to use these software programs.

Figure: KNOPPIX/Math

In regard to our future research and development activities, solving polynomial systems has always been a fundamental and important topic as an application of computational algebra software. Ideal decomposition has already been implemented in Risa/Asir, and can be used to solve polynomial systems. However, ideal decomposition in Risa/Asir has not been updated for a long time. Hence, I would like to rewrite it by incorporating recent developments in order to provide a higher level of performance. Additionally, I would like to employ the numerical/symbolic hybrid computation method of Hirokazu Anai (Fujitsu laboratories and Kyushu University) for computing approximate numerical solutions of polynomial systems.

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