TABLE OF CONTENTS In Part II of this book, we have seen that the numerical algebra of univariate polynomials is governed by a relatively small number of guiding principles: Algebraically, there is the simple structure of a principal ideal, with one generating polynomial which is unique except for a scalar factor, and of a quotient ring and dual space of the same dimension as the degree of that generator, with their multiplicative structure defined by one multiplication matrix. In most cases, the data of a structural description are directly or closely related to the data of the given problem. Therefore, our fundamental idea of describing polynomials with coefficients of limited accuracy by a family of neighborhoods, which embeds numerical algebra into analysis and defines backward errors, could quite directly be put into action in all cases: This made it possible to define the validity of approximate results and to refine approximate results whose accuracy was unsatisfactory. In principle, algorithms for the solution of all meaningful problems in the numerical algebra of univariate polynomials are known; their efficient implementation into software systems is either available or under development.
In the numerical algebra of multivariate polynomials, the situation is very different. We will, at first, consider some of the main reasons for this distinction:
In 
, there is, generally, a wide choice of bases
