Chapter 1 - The characteristic set method and its applications

Abstract

This chapter describes the characteristic set method and its applications. A brief historical review of polynomial equation solving in ancient China and a brief introduction to the characteristic set method, known as Ritt-Wu's, Wu-Ritt's, or Wu's method are discussed. This chapter describes some variants of this general method that incorporate several specialized techniques to enhance the computational performance. This serves to illustrate its applicability and practical efficiency for polynomial equation solving. There are mainly two kinds of methods for polynomial equation solving, which include the numerical ones and symbolic ones. For numerical methods, one may cite the generalized Newton–Raphson method and its diverse variants, and the homotopy continuation method. These methods start directly from the given numerical data in the equations and apply some limiting process to converge toward some solution of the given polynomial equations. In general, these methods will give only one solution among the set of all possible solutions, and are thus local in character in the main. It is found that the ordering of an ascending set will be lowered by adjoining a polynomial reduced with respect to it.