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].

This ERGEBNISSE volume deals with the theory of the minimal description of real constructible sets. It gives a comprehensive state-of-the-art report with many results published for the first time in book form. The authors tried to keep the presentation as self-contained a possible. The book contains numerous examples, historical notes and 63 figures.

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].

This ERGEBNISSE volume deals with the theory of the minimal description of real constructible sets. It gives a comprehensive state-of-the-art report with many results published for the first time in book form. The authors tried to keep the presentation as self-contained a possible. The book contains numerous examples, historical notes and 63 figures.