Investigation of the Dixon Resultants

狄克逊的公式，同时消除几个变量从一个多元多项式方程组将研究。在符号计算中，求解多项式方程和确定方程有共同解的参数条件是一个基本问题，几乎无处不在。多项式被用于许多现象的建模，特别是在机器人、运动学、计算机视觉、实体建模、图形学、化学平衡等工程、CAD-CAM设计等领域。Dixon的公式直接推广了Bezout-Cayley关于消去单个变量的结式公式。Dixon式之所以没有得到广泛的研究，显然是因为它被Macaulay同时消去几个变量的式公式所掩盖。Dixon结果、Macaulay结果和稀疏结果的实现表明Dixon结果在实践中具有更高的计算效率。Dixon结果将被分析，特别强调研究Dixon矩阵的大小和结构与输入多项式系统的关系。＊＊＊

Dixon's formulation for simultaneously eliminating several variables from a system of multivariate polynomial equations will be investigated. Solving polynomials equations and identifying conditions on parameters under which the equations have common solutions are fundamental problems in symbolic computations which arise almost everywhere. Polynomials are used to model many phenomena - particularly in robotics, kinematics, computer vision, solid modeling, graphics, chemical equilibrium and other engineering, CAD-CAM design, etc. Dixon's formulation directly generalizes Bezout-Cayley's formulation of a resultant for eliminating a single variable. Dixon resultants have not been extensively investigated apparently because they presumably got overshadowed by Macaulay's formulation of resultants for simultaneously eliminating several variables. Implementations of Dixon resultants, Macaulay resultants and sparse resultants have revealed that Dixon resultants are more efficient to compute in practice. Dixon resultants will be analyzed with a particular emphasis on studying the relationship of the size and structure of the Dixon matrix to the input polynomial system. ***