TABLE OF CONTENTS In their use as modelling tools in Scientific Computing, polynomials appear, at first, simply as a special class of functions from the 
to 
(or 
to 
). Such polynomials are automatically objects of univariate ( s = 1) or multivariate ( s > 1) analysis over the complex or real numbers. For linear polynomials, this fact has played virtually no role in classical linear algebra; but it has become a fundamental aspect of today's numerical linear algebra where concepts from analysis (norms, neighborhoods, convergence, etc.) and related results are widely used in the design and analysis of computational algorithms. In an analogous manner, the consideration of polynomial algebra as a part of analysis plays a fundamental role in numerical polynomial algebra; it will be widely used throughout this book. In particular, this embedding of algebra into analysis permits the extension of algebraic algorithms to polynomials with coefficients of limited accuracy; cf. Chapter 3.
On the other hand, certain sets of polynomials have special algebraic structures: they may be linear spaces, rings, ideals, etc. Algebraic properties related to these structures may play a crucial role in solving computational tasks involving polynomials, e.g., for finding zeros of polynomial systems; cf. Chapter 2.
In this introductory chapter, we consider various aspects of polynomials which will play a fundamental role...
