Design, Analysis & Applications of Algorithms for Solving Systems of Polynomial Equations & Related Problems

这个项目将探讨几个基本的算法问题 在符号代数计算中， 应用于几何建模。 的总体目标 项目是设计，分析和测试有效的算法 用于求解多项式方程组和相关的 问题，并测试其适用性几何 建模问题。 需要研究的具体问题领域 多项式系统的算法分析 方程，如Wu-Ritt特征集算法和 拉扎德的三角形集算法;发展变化的 这些算法和其他算法来有效地处理 特殊多项式方程;设计有效性 基于Grobner基和一个新基的新算法 多项式理想变换算法; 广义狄克逊结式性质的研究 以及计算它的算法;并探索使用 这些技术在解决几何中的算法问题时 建模 各种算法的设计和分析 将伴随着试验与审判 在现有的计算机代数系统上实现。 目的 实验的一个重要方面就是 分析所提出的改进建议， 分析中可能遗漏的现象。 一些特殊用途算法的微调实现 也在计划中。

This project will explore several basic algorithmic questions in symbolic algebraic computation and investigate their applications to geometric modeling. The overall goal of the project is to design, analyze and test efficient algorithms for solving systems of polynomial equations and related problems, and to test their applicability to geometric modeling problems. The specific problem areas to be studied are: analysis of algorithms for solving system of polynomial equations such as the Wu-Ritt characteristic set algorithm and Lazard's triangular set algorithm; developing variations of these and other algorithms to efficiently handle systems of polynomial equation of special interest; designing efficient new algorithms based on Grobner bases and a new basis conversion algorithm for manipulation of polynomial ideals; investigation of the properties of the general Dixon resultant and algorithms for computing it; and exploration of the use of these techniques in solving algorithmic problems in geometric modeling. The design and analysis of the various algorithms will be accompanied by experimentation with trial implementations on existing computer algebra systems. The aim of the experimentation will be to incorporate possible improvements suggested by the analysis and to uncover phenomena that might possibly be missed out in the analysis. Fine tuned implementations of a few special purpose algorithms are also planned.