Algebraic versus analytic basicness

Abstract

A semialgebraic set is called basic if it can be described by a single system of strict polynomial inequalities. A semianalytic set is called basic if it can be described by a system of strict real analytic inequalities in a neighborhood of each of its points (the system, of course, may depend on the point). In the paper under review the authors describe a solution to the problem of geometric characterization of basic semialgebraic sets among the basic semianalytic sets. This solution appeared in their monograph [Mem. Amer. Math. Soc. 115 (1995), no. 553, vi+117 pp.]. The paper under review is not a summary of this monograph, but rather a complement to it. Instead of outlining the ideas of the proofs, the authors explain the notions of basic sets and their solution to the above problem with the help of many examples. The solution itself is formulated in terms of some collections of ultrafilters on the Boolean algebras of semialgebraic sets and of semianalytic germs; it is "geometric'' compared to the heavy algebraic machinery used in the proofs. The proofs themselves are not discussed in the paper, but the authors hope that their relatively elementary description of the results will motivate the reader to study the relevant algebra.