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Separation of semialgebraic sets


Authors: F. Acquistapace, C. Andradas and F. Broglia
Journal: J. Amer. Math. Soc. 12 (1999), 703-728
MSC (1991): Primary 14P10
DOI: https://doi.org/10.1090/S0894-0347-99-00302-1
Published electronically: April 23, 1999
MathSciNet review: 1672874
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Abstract: In this paper we study the problem of deciding whether two disjoint semialgebraic sets of an algebraic variety over $\mathbb R$ 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.


References [Enhancements On Off] (What's this?)

  • [AcAnBg] F. Acquistapace, C. Andradas, F. Broglia: ``Classification of obstructions for separation of Semialgebraic Sets in dimension 3'', Rev. Matematica U.C.M. 10 (número 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. Bröcker, 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 {\L}ojasiewicz 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: Géométrie algébrique réelle, 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. Bröcker: ``Spaces of Orderings and semialgebraic sets'', Canadian Math. Society conference proc. 4 (1984) 231-248. MR 86m:12002
  • [Br2] L. Bröcker: ``Characterization of fans and hereditarily pythagorean fields'', Math. Z. 151 (1976) 149-163. MR 54:10224
  • [Br3] L. Bröcker: ``On the separation of basic semialgebraic sets by polynomials'' Manuscripta Math. 60 (1988) 497-508 MR 89d:14034
  • [Br4] L. Bröcker: ``On basic semialgebraic sets'', Expo. Math. 9 (1991) 289-334. MR 93b:14085
  • [BrSt] L. Bröcker, 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 field 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: ``Classification of finite 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, Università di Pisa (1998).
  • [Rz] J. Ruiz: ``A note on a separation problem'', Archiv der Mathematik 43 (1984) 422-426. MR 86f:32007

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Additional Information

F. Acquistapace
Affiliation: Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy
Email: acquistf@gauss.dm.unipi.it

C. Andradas
Affiliation: Departamento de Algebra, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Email: andradas@sunal1.mat.ucm.es

F. Broglia
Affiliation: Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy
Email: broglia@gauss.dm.unipi.it

DOI: https://doi.org/10.1090/S0894-0347-99-00302-1
Received by editor(s): February 3, 1997
Received by editor(s) in revised form: August 31, 1998
Published electronically: April 23, 1999
Additional Notes: This work was partially supported by EC contract CHRX-CT94-0506.
The first and third authors are members of GNSAGA of CNR, and were partially supported by MURST
The second author was partially supported by DGICYT PB95-0354 and the Fundación del Amo, UCM
Article copyright: © Copyright 1999 American Mathematical Society

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