Algebraic versus analytic basicness



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Andradas Heranz, Carlos and Ruiz Sancho, Jesús María (1995) Algebraic versus analytic basicness. In Real analytic and algebraic geometry. Walter de Gruyter, Berlin, pp. 1-19. ISBN 3-11-013778-x

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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.

Item Type:Book Section
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Proceedings of the International Conference held in Trento, September 21–25, 1992

Uncontrolled Keywords:Basic semi-analytic set; basic semi-algebraic set; real spectrum; fan
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:21331
Deposited On:13 May 2013 12:34
Last Modified:12 Dec 2018 15:13

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