Effective Semialgebraic Geometry

The first part of the paper introduces the decision problem and the quantifier elimination problem for elementary algebra. Tarski's algorithm, which is based on criteria for the existence of solutions of systems of polynomial equations and inequalities generalizing Sturm's theorem, is presented. Then the text is devoted to the cylindrical algebraic decomposition algorithm of Collins, which relies on the basic methods of the theory of semialgebraic sets initiated by Lojasiewicz. Special attention is paid to the topological aspect of the cylindrical algebraic decomposition, which provides a general method (surely too general to be efficient in practice) for robot motion planning. The techniques and results presented here have no pretention to originality. The references given at the end of the paper are just a sample of the literature on the subject, far from being exhaustive.