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Low dimensional sections of basic semialgebraic sets.

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Andradas Heranz, Carlos and Ruiz Sancho, Jesús María (1994) Low dimensional sections of basic semialgebraic sets. Illinois Journal of Mathematical, 38 (2). pp. 303-326. ISSN 0019-2082

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Abstract

Let X be a real affine algebraic set and S a semialgebraic set. Many important results are known about the basicness of S: mainly, if S is basic open, S can be defined by s strict inequalities, where s is bounded by the dimension of X. It is also known (Br¨ocker-Scheiderer criterion) that
an open semialgebraic set is basic and is defined by s inequalities (s-basic) if and only if for every irreducible subset Y of X the intersection S \ Y is generically s-basic. In a previous paper, the authors proved that to test the basicness of S it is sufficient to test the basicness of the intersections of S with every irreducible surface. In fact, this is the best possible result about the dimension, but it was conjectured by the authors that the first obstruction to s-basicness should be recognized in
dimension s+1.
In the present paper, the authors confirm this conjecture, and in fact they prove the following result: if S is basic, then S is s-basic if and only if for every irreducible subset Y ofX of dimension s+1 the intersection S \Y is generically s-basic.
The proof of this theorem requires a deep analysis of real valuations. The results of independent interest about fans (algebroid fans) defined by means of power series and the approximation problem solved in this case are the main tools of the proof.


Item Type:Article
Uncontrolled Keywords:basic open semi-algebraic set; fans; real spectrum
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:14809
Deposited On:18 Apr 2012 08:47
Last Modified:09 Aug 2018 08:04

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