“The development of polynomial-elimination techniques from classical theory to modern algorithms has undergone a tortuous and rugged path. This can be observed from B.L. van der Waerden’s elimination of the elimination theory chapter from his classic Modern Algebra in later editions, A. Weil’s hope to eliminate ’from algebraic geometry the last traces of elimination theory’, and S. Abhyankar’s suggestion to ’eliminate the eliminators of elimination theory’. ...”
(From the preface).
In fact, the problem of elimination has, historically, initiated algebraic geometry. In my view, the full difficulty and depth of the problem has only been recognized in the last few decades, as the algorithmic aspect of the problem has seen a renaissance. This book, therefore, is not only topical, but is also the first book that contains all major currently available algorithmic approaches on elimination theory in one conceptional frame.
After an introductory chapter that compiles the prerequisites, chapter 2 reports on zero decomposition of polynomial systems based on the characteristic sets approach. Chapter 3 describes zero decomposition into simple systems with projection, and chapter 4 considers decomposition into irreducible simple systems. These three chapters constitute the main body of the book, and report mainly on the author’s own research in the tradition of W.T. Wu and of people in this community. Chapter 5 summarizes the other current algorithmic approaches to elimination theory mainly based on Groebner bases theory and resultants. Chapter 6 provides some of the important applications of elimination theory in algebraic geometry and polynomial ideal theory, notably to algorithmic decomposition of algebraic varieties. Chapter 7, finally, gives a few applications of algorithmic eliminations, in particular automated geometrical theorem proving.
The book is well suited both as a textbook and as a research reference for the state of the art in algorithmic elimination theory. Theorems and proofs are presented in such a way that the correctness of the algorithms based on them can easily be verified. Well-chosen examples make it easy to understand the concepts and the methods. The main contents of this book - although not all details - should enter standard curricula on algebra.
In my opinion, a future book on the subject could better bridge the gap between the formal language in which theorems and proofs are given and the formal language in which algorithms are presented. Also, a better balance between the author’s own approach and the alternative approaches might be worthwhile.