Memoirs of the American Mathematical Society 1995; 117 pp; softcover Volume: 115 ISBN10: 0821826123 ISBN13: 9780821826126 List Price: US$41 Individual Members: US$24.60 Institutional Members: US$32.80 Order Code: MEMO/115/553
 A set which can be defined by systems of polynomial inequalities is called semialgebraic. When such a description is possible locally around every point, by means of analytic inequalities varying with the point, the set is called semianalytic. If one single system of strict inequalities is enough, either globally or locally at every point, the set is called basic. The topic of this work is the relationship between these two notions. Namely, Andradas and Ruiz describe and characterize, both algebraically and geometrically, the obstructions for a basic semianalytic set to be basic semialgebraic. Then they describe a special family of obstructions that suffices to recognize whether or not a basic semianalytic set is basic semialgebraic. Finally, they use the preceding results to discuss the effect on basicness of birational transformations. Readership Advanced graduate students and specialists in real algebra and real geometry. Table of Contents  Introduction
 Basic and generally basic sets
 The real spectrum
 Algebraic and analytic tilde operators
 Fans and basic sets
 Algebraic fans and analytic fans
 Prime cones and valuations
 Centers of an algebraic fan
 Henselization of algebraic fans
 A goingdown theorem for fans
 Extension of real valuation rings to the henselization
 The amalgamation property
 Algebraic characterization of analytic fans
 Finite coverings associated to a fan
 Geometric characterization of analytic fans
 The fan approximation lemma
 Analyticity and approximation
 Analyticity after birational blowingdown
 References
