## Separation of semialgebraic sets

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- by F. Acquistapace, C. Andradas and F. Broglia
- J. Amer. Math. Soc.
**12**(1999), 703-728 - DOI: https://doi.org/10.1090/S0894-0347-99-00302-1
- Published electronically: April 23, 1999
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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

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## Bibliographic 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
- MR Author ID: 41870
- Email: broglia@gauss.dm.unipi.it
- 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 - © Copyright 1999 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**12**(1999), 703-728 - MSC (1991): Primary 14P10
- DOI: https://doi.org/10.1090/S0894-0347-99-00302-1
- MathSciNet review: 1672874