RT Journal Article
T1 Low dimensional sections of basic semialgebraic sets.
A1 Andradas Heranz, Carlos
A1 Ruiz Sancho, Jesús María
AB 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) thatan 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 indimension 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.
PB Univ Illinois Urbana-Champagn
SN 0019-2082
YR 1994
FD 1994
LK https://hdl.handle.net/20.500.14352/57159
UL https://hdl.handle.net/20.500.14352/57159
LA eng
NO DGICYT.
DS Docta Complutense
RD 6 abr 2025