Computer Algebra Seminar No.19 (計算代数セミナー, 第19回) at Kyushu Univ.(九州大学)

November 5 (Wed), 2003. 11月5日 (水), 13:00 --- 17:30

場所: 九州大学理学部 (箱崎キャンパス) 3 号館 3311 教室


Tadashi Takahashi (Kobe University)
An Application of Elimination Ideal for a Defining Equation of Singularity , 13:00-14:00

We can obtain non-degeneracy conditions of singularities by using Gr\"obner basis. We can treat divisions of polynomials that have parameter coefficients and indeterminate exponents by using pseudo-remainders. In classification of singularities using hierarchy, the defining equations are polynomials with parameter coefficients and indeterminate exponents.
In general, the forms of Gr\"obner bases of ideals generated by such polynomials depend on the values of parameters, and those calculations are not easy. We will see that a stratification problem of coefficients and exponents of a set of polynomials is naturally raised to classify singularities. We will give a partial answer to this problem.

Volker Weispfenning (Passau University)
Canonical Comprehensive Groebner Bases (Tentative), 14:10--15:40

Hirokazu Anai (FUJITSU Laboratories)
Quantifier Elimination for Real Algebraic Constraints , 15:50--16:40

The wide range of problems in science and engineering can be reduced to that of solving real algebraic constraints. Hence, making progress in solving real algebraic constraints has a significant impact on those areas. Quantifier elimination (QE) provides exact solutions for real algebraic constraints and makes it possible to deal with the constraints parametrically. These are significant advantages of QE in solving such constraints and lead to a way of resolving difficult problems (e.g. nonconvex cases) that are hard to solve with numerical methods. Thus, QE has been applied to various kinds of problems in science and engineering. Though the actual applicability of QE has been fairly limited due to the practical complexity of the implemented QE methods, some of these methods have been able to solve the problems of interesting size in application fields by virtue of the enormous increase in computational power and theoretical work. In this talk, we show the successful applications of some QE methods to industrial problems with concrete examples from system and control theory. Then, we present some promising directions to achieve efficiency of the methods solving real algebraic constraints by combining the QE based approach with the numerical approach.

Hitoshi Yanami (FUJITSU Laboratories)
SyNRAC: Maple 上の代数制約問題解決ツールボックス , 16:40--17:30

理工学や産業上の課題を解決する手段として「記号・代数計算技術」が注目 を浴びるようになってきた。従来の数値的計算を前提としたシミュレーション 技術とは異なり、記号・代数計算技術は不定元やパラメタなどの記号が入った 式を多項式のまま扱うことに特徴がある。多項式を計算機で取り扱う場合、一 般には処理時間が掛かり扱える問題にも制約があるが、数値的計算技術にはな い利点もある。例えば、1)非線形や非凸な制約問題を取り扱うことができる、 2)パラメタを記号としてそのまま扱うため、設計仕様を満たす集合をパラメ タ空間内の領域として正確に求めることができる、などである。 現在、我々の研究グループでは、記号・代数計算技術を用いたツールボック ス SyNRAC を Maple 上で開発している。本発表では SyNRAC 開発の現状を限 定子除去法の実装を中心に報告する。

Past Lectures.

The computer algebra seminar is co-organized by
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