Masayuki Noro

Risa/Asir—a computer algebra system

A computer algebra system risa/asir consists of a command asir for interactive use and several subroutine libraries which can be used as the parts of other programs. The grammar of the user language of asir is a variant of that of C and asir has a built-in dbx-like debugger. Risa’s subroutine libraries include basic arithmetic subroutines, parser, evaluator and storage manager, and each of them can be used individually. This paper describes the characteristics and structure of risa system. We also show some sample programs, usage of the debugger and some timing data of fundamental calculations.

Holonomic gradient descent and its application to the Fisher-Bingham integral

We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere

A Modular Method to Compute the Rational Univariate Representation of Zero-dimensional Ideals

S^n

A Computer Algebra System: Risa/Asir

, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.

Efficient Implementation of Schoof's Algorithm

To give an efficiently computable representation of the zeros of a zero-dimensional ideal I, Rouillier (1996) introduced the rational univariate representation (RUR) as an extension of the generalized shape lemma (GSL) proposed by Alonso et al. (1996). In this paper, we propose a new method to compute the RUR of the radical of I, and report on its practical implementation. In the new method, the RUR of the radical of I is computed efficiently by applying modular techniques to solving the systems of linear equations. The performance of the method is examined by practical experiments. We also discuss its theoretical efficiency.

Computation of the splitting fields and the Galois groups of polynomials

Risa/Asir [20] is a computer algebra system developed for efficient algebraic computation. It is also designed to be a platform for parallel distributed computation under the OpenXM (Open Message eXchange for Mathematics) protocol [11]. OpenXM defines client-server communication between mathematical software systems and Risa/Asir is one of main components in the OpenXM package [21]. The source code of Risa/Asir is completely open and it is easy to modify the source code or to add new functions. In the present paper, we explain an overview of Risa/Asir, its functions and implemented algorithms with their performances, the OpenXM API and the way to add built-in functions.

Implementation of prime decomposition of polynomial ideals over small finite fields

Schoofs algorithm is used to find a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkiess method, the isogeny cycles method, and trial search by match-and-sort techniques, we can count the number of rational points on an elliptic curve over GF(p) in a reasonable time, where p is a prime whose size is around 240-bits.

On a Conjecture for the Dimension of the Space of the Multiple Zeta Values

This study is a continuation of Yokoyama et al. [22], which improved the method by Landau and Miller [11] for the determination of solvability of a polynomial over the integers. In both methods, the solvability of a polynomial is reduced, in polynomial time, to that of polynomials, each of which is constructed so that its Galois group acts primitively on its roots. Then, by virtue of Palfy’s bound [14], solvability of polynomials with primitive Galois groups can be determined in polynomial time. An effective method, thus, exists in theory. For practical computation, however, the most serious problem remains: How to determine solvability of each polynomial with primitive Galois group.

Modular dynamic evaluation

Abstract An algorithm for the prime decomposition of polynomial ideals over small finite fields is proposed and implemented on the basis of previous work of the second author. To achieve better performance, several improvements are added to the existing algorithm, with strategies for computational flow proposed, based on experimental results. The practicality of the algorithm is examined by testing the implementation experimentally, which also reveals information about the quality of the implementation.

Algorithms for computing a primary ideal decomposition without producing intermediate redundant components

Since Euler, values of various zeta functions have long attracted a lot of mathematicians. In computer algebra community, Apery’s proof of the irrationality of ζ(3) is well known. In this paper, we are concerned with the “multiple zeta value (MZV)”. More than fifteen years ago, D. Zagier gave a conjecture on MZVs based on numerical computations on PARI. Since then there have been various derived conjectures and two kinds of efforts for attacking them: one is a mathematical proof and another one is a computational experiment to get more confidence to verify a conjecture. We have checked one of these conjectures up to weight k = 20, which will be explained later, with Risa/Asir function for non-commutative polynomials and special parallel programs of linear algebra designed for this purpose.