Jesus M. Ruiz

The plan to write this book was laid out in April 1987 at Oberwolfach, dur- ing the Conference "Reelle Algebraische Geometrie". Afterwards we met at various conferences and seminars in Luminy, Madrid, Munster, Oberwolfach, Segovia, Soesterberg, Trento and La Turballe. We would like to thank the or- ganizers and the institutions which supported these meetings. With pleasure we remember the special year on Real Algebraic Geometry and Quadratic Forms (Ragsquad) in Berkeley 1990/91 where an essential part of this book was written. Thanks to T.Y. Lam and R. Robson. We are indepted to our Departments: Universidad Complutense de Madrid and WestfiiJische Wilhelms-Universitiit Munster, as well as the D.A.A.D. and the D.G.I.C.y'T .. It is not possible to mention here all colleagues and friends who showed permanent interest in the project. They encouraged us to continue and com- plete this work. In particular, we are obliged to Jacek Bochnak, Mike Buchner, Michel Coste and Claus Scheiderer for proofreading and helpful suggestions. While the work was still in progress, Manfred Knebusch used parts of our manuscript for a course on Real Algebraic Geometry. His experience con- vinced us that it was worth pursuing a fully abstract approach. We were also in permanent contact with Murray Marshall, and the reader will recognize the mutual influence of ideas. We are also indebted to Professor Reinhold Remmert and to the Springer- Verlag for publishing the book in this series.

The aim of these notes is to cover the basic algebraic tools and results behind the scenes in the foundations of Real and Complex Analytic Geometry. The author has learned the subject through the works of many mathematicians, to all of whom he is indebted. However, as the reader will immediately realize, he was specially influenced by the writings of S.S. Abhyankar and J .-C. Tougeron. In any case, the presentation of all topics is always as elementary as it can possibly be, even at the cost of making some arguments longer. The background formally assumed consists of: 1) Polynomials: roots, factorization, discriminant; real roots, Sturm's Theorem, formally real fields; finite field extensions, Primitive Element Theorem. 2) Ideals and modules: prime and maximal ideals; Nakayama's Lemma; localiza- tion. 3) Integral dependence: finite ring extensions and going-up. 4) Noetherian rings: primary decomposition, associated primes, Krull's Theorem. 5) Krull dimension: chains of prime ideals, systems of parameters; regular systems of parameters, regular rings. These topics are covered in most texts on Algebra and/or Commutative Algebra. Among them we choose here as general reference the following two: - M. Atiyah, I.G. Macdonald: Introduction to Commutative Algebra, 1969, Addison-Wesley: Massachusetts; quoted A-McD] . - S. Lang: Algebra, 1965, Addison-Wesley: Massachusetts; quoted L].