Real Algebraic Varieties

This book gives a systematic presentation of real algebraic varieties. Real algebraic varieties are ubiquitous.They are the first objects encountered during the learning of coordinates then equations, but the systematic study of these objects, however elementary they may be, is formidable. The book is intended for two kinds of audiences; It accompanies the reader familiar with algebra and geometry at the master level, in learning the basics of this rich theory, as much as it brings to the most advanced reader many fundamental results often missing from the available literature, the «folklore». In particular, the introduction of topological methods of the theory to non-specialists is one of the original features of the book. The first three chapters introduce the basics and classical methods of real and complex algebraic geometry. The last three chapters each focus on one more specific aspect of real algebraic varieties. A panorama of classical knowledge is presented, as well as major developments of the last twenty years in terms of topology and geometry of varieties of dimension two and three, without forgetting the curves, the central subject of Hilbert's famous sixteenth problem. Exercises of various levels are given, and the solutions of many of them are provided at the end of each chapter.