“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.