After the introduction of coordinates in geometry by Descartes and Fermat, it seemed that all questions in geometry could be reduced to algebra. As Davis and Hersh say in Descartes' Dream, one hoped that this would eliminate the need for mysterious insights that many times are needed to solve seemingly elementary problems in plane geometry — those insights would be replaced by algebraic manipulation.
While Hilbert's Nullstellensatz provides an explicit correspondence between geometric objects (algebraic varieties) and sets of polynomials (radical ideals) and Tarski's elimination theory for real closed fields shows that algebraic formulations of geometric problems is a decidable problem, algebraic insights are still needed to solve non-trivial problems in analytic geometry. Tarski's result does not yield a practical algorithm.
All this changed recently with the work of Wu on triangulating polynomial systems and Buchberger's algorithm for computing Gröbner bases of polynomial ideals. It is now possible in many cases to decide reliably and rather quickly whether a polynomial representing a geometric statement is in the ideal generated by the hypotheses, thus proving or disproving the geometric statement.
Pech's book shows many examples of this method in action. An interesting feature of the book is that it includes and compares mechanical proofs generated by CoCoA or SINGULAR and classical synthetic proofs for the same problems.
Besides the instructive contrast between mechanical proofs and synthetic proofs, the book also includes much interesting geometry. However, it is not meant as a textbook. Instructors interested in using it for a course will have to complement it with material on algorithms from the books by Cox, Little, and O'Shea (Using Algebraic Geometry and Ideals, Varieties, and Algorithms), as recommended by Pech.
Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.