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Semi-algebraic geometry with rational continuous functions

  • Jean-Philippe Monnier
Article

Abstract

Let X be a real algebraic subset of \({{\mathrm{\mathbb {R}}}}^n\). We investigate the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.

Mathematics Subject Classification

14P99 11E25 26C15 

Notes

Acknowledgements

I want to thank G. Fichou and R. Quarez for stimulating conversations concerning regulous functions. I thank also F. Mangolte and D. Naie for interesting and helpful discussions. I thank a referee of an earlier version of the paper for pointing out to us a mistake in this earlier version. In memory of J.-J. Risler.

References

  1. 1.
    Andradas, C., Bröcker, L., Ruiz, J.M.: Constructible Sets in Real Geometry. Springer (1996)Google Scholar
  2. 2.
    Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)zbMATHGoogle Scholar
  3. 3.
    Bierstone, E., Milman, P.D.: Arc-analytic functions. Invent. Math. 101, 411–424 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bochnak, J., Coste, M., Roy, M.-F.: Géométrie algébrique réelle. Springer (1987)Google Scholar
  5. 5.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer (1998)Google Scholar
  6. 6.
    Bonnard, I.: Un critère pour reconnaître les fonctions algébriquement constructibles. J. Reine angew. Math. 526, 61–88 (2000)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bonnard, I.: Description of algebraically constructible functions. Adv. Geom. 3, 145–161 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bonnard, I.: Nash constructible functions. Manuscr. Math. 112, 55–75 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bröcker, L.: On basic semialgebraic sets. Expos. Math. 9(4), 289–334 (1991)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Coste, M.: Real algebraic sets in Arc spaces and additive invariants in real algebraic and analytic geometry. Panor. Synthèses. Soc. Math. France 24, 1–32 (2007)Google Scholar
  11. 11.
    Kollár, J., Nowak, K.: Continuous rational functions on real and p-adic varieties. Math. Z. 279(1–2), 85–97 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kollár, J., Kucharz, W., Kurdyka, K.: Curve-rational functions. Math. Ann. (2017).  https://doi.org/10.1007/s00208-016-1513-z zbMATHGoogle Scholar
  13. 13.
    Kucharz, W.: Rational maps in real algebraic geometry. Adv. Geom. 9(4), 517–539 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kucharz, W.: Piecewise-regular map. Math. Ann. (2017).  https://doi.org/10.1007/s00208-017-1607-2 Google Scholar
  15. 15.
    Kucharz, W., Kurdyka, K.: Stratified-algebraic vector bundles. J. Reine Angew. Math. (2016).  https://doi.org/10.1515/crelle-2015-0105 zbMATHGoogle Scholar
  16. 16.
    Kurdyka, K.: Ensembles semialgébriques symétriques par arcs. Math. Ann. 282, 445–462 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kurdyka, K., Parusiński, A.: Arc-symmetric sets and arc-analytic mappings in Arc spaces and additive invariants in real algebraic and analytic geometry. Panor. Synthèses. Soc. Math. Fr. 24, 33–67 (2007)zbMATHGoogle Scholar
  18. 18.
    Fichou, G., Huisman, J., Mangolte, F., Monnier, J.-P.: Fonctions régulues. J. Reine angew. Math. 718, 103–151 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fichou, G., Monnier, J.-P., Quarez, R.: Continuous functions on the plane regular after one blowing-up. Math. Z. 285, 287–323 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    McCrory, C., Parusiński, A.: Algebraically constructible functions. Ann. Sci. Ecole Norm. Sup. 4 série 30, 527–552 (1997)MathSciNetzbMATHGoogle Scholar
  21. 21.
    McCrory, C., Parusiński, A.: Topology of real algebraic sets of dimension \(4\): necessary conditions. Topology 39, 495–523 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Parusiński, A., Szafraniec, Z.: Algebraically constructible functions and signs of polynomials. Manuscr. Math. 93, 443–456 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shafarevich, I.R.: Basic Algebraic Geometry. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LAREMAAngersFrance
  2. 2.Université d’AngersAngersFrance

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