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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Separation of semialgebraic sets
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by F. Acquistapace, C. Andradas and F. Broglia PDF
J. Amer. Math. Soc. 12 (1999), 703-728 Request permission

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.
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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
  • 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