Complutense University Library

Separation of semialgebraic sets

Acquistapace, Francesca and Andradas Heranz, Carlos and Broglia, Fabrizio (1999) Separation of semialgebraic sets. Journal of the American Mathematical Society, 12 (3). pp. 703-728. ISSN 0894-0347

[img] PDF
Restricted to Repository staff only until 2020.

424kB

Official URL: http://www.ams.org/jourcgi/jrnl_toolbar_nav/jams_all

View download statistics for this eprint

==>>> Export to other formats

Abstract

We study the problem of deciding whether two disjoint semialgebraic sets of an algebraic variety over R are separable by a polynomial. For that we isolate a dense
subfamily of spaces of orderings, named geometric, which suffice to test separation and that reduce the problem to the study of the behaviour of the semialgebraic sets in their boundary. Then we derive several characterizations for the generic separation, among which there is a geometric criterion that can be tested algorithmically. Finally we show how to check recursively whether we can pass from generic separation to separation,
obtaining a decision procedure for solving the problem.

Item Type:Article
Uncontrolled Keywords:decidability of separation problem of semialgebraic sets; algorithm
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:14799
References:

[AcAnBg] F. Acquistapace, C. Andradas, F. Broglia: \Classication of obstructions for separation

of Semialgebraic Sets in dimension 3", Rev. Matematica U.C.M. 10 (numero

suplementario) (1997) 27{49. CMP 98:05

[AcBgFo] F. Acquistapace, F. Broglia, E. Fortuna: \A separation theorem in dimension 3",

Nagoya Math. Journal 143 (1996) 171{193. MR 97k:14056

[AnBrRz] C. Andradas, L. Br¨ocker, J.M. Ruiz: Constructible sets in real geometry, Ergeb. Math.

33, Springer-Verlag Berlin-Heidelberg-New York, 1996. MR 98e:14056

[AnRz1] C. Andradas, J.M. Ruiz: \More on basic semialgebraic sets", in Real Algebraic and

analytic geometry, Lect. Notes in Math. 1524, Springer-Verlag, New York, (1992),

128-139. MR 94g:14030

[AnRz2] C. Andradas, J.M. Ruiz: \Low dimensional sections of basic semialgebraic sets", Illinois

J. of Math. 38 (1994), 303-326. MR 95d:14056

[AnRz3] C. Andradas, J.M. Ruiz: \Ubiquity of Lojasiewicz example of a non-basic semialgebraic

set", Michigan Math. J. 41 (1994). MR 96e:14064

[BeNe] E. Becker, R. Neuhaus: \Computation of real radicals of polynomial ideals", Proc.

MEGA 92, Nice, France, Birkhauser (1993) 1{20. MR 94g:12001

[BiMi] E. Bierstone, P. Milman: \Canonical desingularization in characteristic zero by

blowing-up the maximum strata of a local invariant", Inventiones Math. 128 (1997),

207-302. MR 98e:14010

[BCR] J. Bochnak, M. Coste, M.-F. Roy: Geometrie algebrique reelle, Ergeb. Math. 12,

Springer-Verlag, Berlin-Heidelberg-New York, 1987. MR 90b:14030

[BoEf] J. Bochnak, G. Efroymson: \Real Algebraic geometry and the Hilbert 17th problem".

Math. Ann. 251 (1980) 213{241. MR 81k:14023

[Br1] L. Br¨ocker: \Spaces of Orderings and semialgebraic sets", Canadian Math. Society

conference proc. 4 (1984) 231{248. MR 86m:12002

[Br2] L. Br¨ocker: \Characterization of fans and hereditarily pythagorean elds", Math. Z.

151 (1976) 149{163. MR 54:10224

[Br3] L. Br¨ocker: \On the separation of basic semialgebraic sets by polynomials"

Manuscripta Math. 60 (1988) 497{508 MR 89d:14034

[Br4] L. Br¨ocker: \On basic semialgebraic sets", Expo. Math. 9 (1991) 289-334. MR

93b:14085

[BrSt] L. Br¨ocker, G. Stengle: \On the Mostowski number", Math. Z. 203 (1990) 629{633.

MR 91g:14058

[He] G. Hermann: \Die Frage der endlich vielen Schritte in der Theorie der Polynomideale",

Math. Annalen 95 (1926).

[Hk] H. Hironaka: \Resolution of singularities of an algebraic variety over a eld of characteristic

zero", Annals of Math. 79 (1964) I:109-123, II:205-326. MR 33:7333

[Mo] T. Mostowski: \Some properties of the ring of Nash functions", Ann. Scuola Norm.

Sup. Pisa 3 (1976) 245{266. MR 54:307

[Mr1] M. Marshall: \Classication of nite spaces of orderings", Canad. J. Math. 31 (1979)

320-330. MR 80i:10026

[Mr2] M. Marshall: \Quotients and inverse limits of spaces of orderings", Canad. J. Math.

31 (1979) 604-616. MR 80f:10021

[Mr3] M. Marshall: \The Witt ring of a space of orderings", Trans. Amer. Math. Soc. 258

(1980) 505-521. MR 81b:10012

[Mr4] M. Marshall: \Spaces of orderings IV", Canad. J. Math. 32 (1980) 603-627. MR

81m:10035

[Mr5] M. Marshall: \Spaces of orderings and Abstract Real Spectra", Lect. Notes in Math.

1636, Springer-Verlag, New York, (1997). MR 98b:14041

[Ne] R. Neuhaus: \Computation of real radicals of polynomial ideals II", Proc. MEGA 92,

Nice, France, Birkhauser (1993). MR 94g:12001

[Pr] A. Prestel: \Model Theory for the Real Algebraic Geometer", to appear as a Quaderni

del Dottorato del Dipartimento de Matematica, Universita di Pisa (1998).

[Rz] J. Ruiz: \A note on a separation problem", Archiv der Mathematik 43 (1984) 422-426.

Deposited On:18 Apr 2012 08:32
Last Modified:06 Feb 2014 10:09

Repository Staff Only: item control page