Development of efficient algorithms and software for solving parametric systems

这项研究的目的是开发高效的算法和软件来求解多项式系统或含参数的微分方程组。研究结果如下：1.动态求值允许我们用不可约定义多项式来定义代数数。我们利用理想商的概念对动态评价进行了重新定义，并提出了一种模块化的方法来改进计算。将新方法应用于离散综合Groebner基的计算。提出了一种计算综合Groebner基(CGB)和综合Groebner系(CGS)的新方法。该方法可以在一般多项式环上用Groebner基计算来实现，并且改进了CGB或CGS.3的计算量。研究了带参数的多项式理想的结构。特别地，我们重点研究了参数多项式理想的稳定性，并在简单情况下发现了这类理想的周期性和渐近性。给出了有理函数域或幂等级数环上微分算子环的余数除法的算法。我们构造了A-超几何系统的收敛解。利用高斯超几何函数的差分方程组，导出了它们的二次关系。为了计算这些对象，我们开发了一个名为‘Yang’的软件包，用于计算有理函数域上的差-微分算子环。所有上述结果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得。在日本数学学会的会议上，我们举办了名为“数学软件和免费文档”的研讨会。此外，我们构建了一个运行KNOPPIX/Math的虚拟机，并分发了包含该虚拟机的DVD。

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.