Complutense University Library

Low dimensional sections of basic semialgebraic sets.


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

[img] PDF
Restringido a Repository staff only hasta 2020.


Official URL:


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

C. ANDRADAS, Specialization chains of real valuation rings, J. Algebra 124 (1989), 437-446.

C. ANDRADAS, L. BRCKER and J. M. RuIz, Minimal generation of basic open semianalytic sets, Invent. math. 92 (1988), 409-430.

Real algebra and analytic geometry to appear.

C. ANDRADAS and J. M. RuIz, "More on basic semialgebraic sets" in Real algebraic and analytic geometry, Lecture Notes in Math., no. 1524, Springer-Verlag, New

York, 1992, pp. 128-139.

On local uniformization of orderings, Contemp. Math., to appear.

Algebraic fans versus analytic fans, Mem. Amer. Math. Soc., to appear.

J. BOCHNAK, M. COSTE and M.-F. Roy, Gdomdtrie algdbrique rdelle, Ergeb. Math.,

vol. 12, Springer-Verlag, New York, 1987.

[BR1] L. BR6CKER, Characterization of fans and hereditarily pythagorean fields, Math.

Zeitschr. 151 (1976), 149-163.

Minimale Erzeugung yon Positivbereichen, Geom. Dedicata 16 (1984), 335-350.

"On the stability index of noetherian rings" in Real analytic and algebraic

geometry, Lecture Notes in Math., no. 1420, Springer-Verlag, New York, 1990, pp. 72-80.

On basic semialgebraic sets, Expo. Math. 9 (1991), 289-334.

L. BRtSCKER and H. W. SCHOLTING, Valuation theory from the geometric point of

view, J. Reine Angew. Math. 365 (1986), 12-32.

n. HIRONAKA, Resolution of singularities of an algebraic variety over a field of

characteristic zero, Ann. of Math. 79 (1964), 109-123, 205-326.

J.-P. JOUANOLOU, Thdormes de Bertini et applications, Progress in Math., no. 42, Birkhiiuser, Boston, 1983.

L. MAHI, Une ddmostration dldmentaire du thdorme de Brfcker-Scheiderer, C. R. Acad. Sci. Paris Serie 309 (1989), 613-616.

M. MARSHALL, Classification offinite spaces of orderings, Canad. J. Math. 31 (1979) 320-330.

Quotients and inverse limits of spaces of orderings, Canad. J. Math. 31

(1979), 604-616.

The Witt ring of a space of orderings, Trans. Amer. Math. Soc. 298 (1980),


Spaces of orderings IV, Canad. J. Math. 32 (1980), 603-627.

Spaces of orderings: systems of quadratic forms, local structure and saturation, Comm. Algebra 1 (1984), 723-743.

Minimal generation of basic sets in the real spectrum of a commutative ring, to appear.

R. ROBSON, Nash wings and real prime divisors, Math. Ann. 273 (1986), 177-190.

J. M. Ruiz, Cnes locaux et completions, C. R. Acad. Sc. Paris (I) 302 (1986), 177-190.

On’ the real spectrum of a ring of global analytic functions, Publ. Inst. Recherche Math. Rennes 4 (1986), 84-95.

J. M. Ruiz and M. SHIOTA, On global Nash functions, Ann. Sci. tcole Norm. Sup., to appear.

C. SCHEIDERER, Stability index of real varieties, Invent. Math. 97 (1989), 467-483.

J.-C. TOUGERON, ldd.altx de fonctions diffHentiables, Ergeb. Math., no. 71, Springer-Verlag, New York, 1972.

Deposited On:18 Apr 2012 08:47
Last Modified:06 Feb 2014 10:09

Repository Staff Only: item control page