Development of efficient algorithms and software for solving parametric systems

开发用于求解参数系统的高效算法和软件

• 批准号: 17340028
• 负责人: NORO Masayuki
• 金额: $ 9.15万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340028/
• 关键词: Groebner Basis; F4 algorithm; Algebraic extension; CGB; Mathematical Software; Hvnereeometric equation; polynomial system; modular computation; F_4; 斉次化; 数式処理; 計算代数; パラメタ; dynamic evaluation

The purpose of this research is to develop efficient algorithms and software for solving polynomial systems or system of differential equations containing parameters. The research results are as follows:1. Dynamic evaluation allows us to define algebraic numbers by non-irreducible defining polynomial. We reformulated the dynamic evaluation by using the notion of ideal quotient, and we proposed a modular method to improve the computation. We applied the new method for the computation of discrete comprehensive Groebner basis.2. We proposed a new method for computing comprehensive Groebner basis (CGB) and comprehensive Groebner system (CGS). This method can be implemented by using Groebner basis computation over usual polynomial ring and it improves the computation of CGB or CGS.3. We investigated the structures of polynomial ideals with parameters. In particular, we focused on the stability of parametric polynomial ideals and we found a periodicity and asymptotic behavior of such ideals in simple cases.4. We proposed algorithms for doing division with remainder in rings of differential operators over the field of rational functions or the power series rings. We constructed convergent solutions of A-hypergeometric systems. By using the system of difference equations for Gauss hypergeometric functions, we derived their quadratic relations. For computing these objects we developed a software package named ‘yang' for computing in rings of difference-differential operators over the field of rational functions.5. All the results above have been implemented in a computer algebra system Risa/Asir, which is available from our web site. In the conferences of Japan Mathematical Society, we held workshops named "Mathematical software and free documents". Furthermore, we constructed a virtual machine on which KNOPPIX/Math runs and we distributed DVDs containing the virtual machine.

本研究的目的是开发有效的算法和软件来求解多项式系统或包含参数的微分方程组。研究结果如下： 1．动态求值允许我们通过不可约定义多项式来定义代数数。我们使用理想商的概念重新表述了动态评估，并提出了一种模块化方法来改进计算。应用新方法进行离散综合Groebner基的计算。 2．我们提出了一种计算综合格罗布纳基（CGB）和综合格罗布纳系统（CGS）的新方法。该方法可以通过在普通多项式环上使用Groebner基计算来实现，并且改进了CGB或CGS.3的计算。我们研究了带参数的多项式理想的结构。特别地，我们关注了参数多项式理想的稳定性，并发现了这种理想在简单情况下的周期性和渐近行为。 4．我们提出了在有理函数域或幂级数环上的微分算子环中进行余数除法的算法。我们构建了 A 超几何系统的收敛解。通过使用高斯超几何函数的差分方程组，我们推导了它们的二次关系。为了计算这些对象，我们开发了一个名为“yang”的软件包，用于在有理函数领域的差分微分算子环中进行计算。 5．上述所有结果均已在计算机代数系统 Risa/Asir 中实现，该系统可从我们的网站获取。在日本数学会的会议上，我们举办了名为“数学软件和自由文档”的研讨会。此外，我们还构建了一个运行 KNOPPIX/Math 的虚拟机，并分发了包含该虚拟机的 DVD。

项目成果

The Buchberger algorithm in the ring of differential operators and its applications

微分算子环中的Buchberger算法及其应用

• 发表时间: 2006
• 作者: Nobuki;Takayama
• 通讯作者: Takayama

Computation of Full Comprehensive Groebner Bases

完全综合 Groebner 基的计算

• 发表时间: 2005
• 期刊: Lecture Notes in Computer Science 3718
• 作者: M. Granger;T. Oaku;N. Takayama;S.Iwamoto;Y.Kawano;Y.Yoshida;木村 欣司;高山 信毅;H.Anai;S.Iwamoto;A.Suzuki
• 通讯作者: A.Suzuki

An efficient implementation for computing groebner bases over algebraic number fields

计算代数数域上的格罗布纳基的有效实现

• 发表时间: 2006
• 期刊: Proceedings of ICMS 2006, LNCS, Springer 4151
• 作者: Y. Kurata;M. Noro;Y.Yoshida;M.Noro;Y.Yoshida;R.Bahloul;Y.Yoshida;M. Noro
• 通讯作者: M. Noro

Local Grobner Fan

当地格罗布纳粉丝

• 发表时间: 2007
• 期刊: C.R. Acad.Sci. Paris, Ser. I. (印刷中)
• 作者: Y. Kurata;M. Noro;Y.Yoshida;M.Noro;Y.Yoshida;R.Bahloul
• 通讯作者: R.Bahloul

A Simple Algorithm to compute Comprehensive Groebner Bases using Groebner Bases

使用 Groebner 基计算综合 Groebner 基的简单算法

• 发表时间: 2006
• 期刊: Proc. ISSAC2006(ACM Press)
• 作者: A. Suzuki;Y. Sato
• 通讯作者: Y. Sato