Hausdorff Discretizations of Algebraic Sets and Diophantine Sets

  • Mohamed Tajine
  • Christian Ronse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)

Abstract

This paper is a continuation of our works [12],[13],[15],[16],[17],[18] [21]

We study the properties of Hausdorff discretizations of algebraic sets. Actually we give some decidable and undecidable properties concerning Hausdorff discretizations of algebraic sets and we prove that some Hausdorff discretizations of algebraic sets are diophantine sets. We refine the last results for algebraic curves and more precisely for straight lines.

Keywords

Algebraic set diophantine set Hausdorff discretization homogeneous metric 

References

  1. 1.
    E. Andres. Standard Cover: a new class of discrete primitives. Internal Report, IRCOM, Université de Poitiers. 100, 107Google Scholar
  2. 2.
    J. Bochnak, M. Coste, M.-F. Roy. Real Algebraic Geometry. Serie of Modern Surveys in Mathematics, Vol. 36, Springer, 1998. 105Google Scholar
  3. 3.
    H. Busemann The geometry of geodesics. Academec Press, New York, 1955. 100, 101MATHGoogle Scholar
  4. 4.
    J. B. J. Fourier. Solution d’une question particuliére du calcul des inégalités. Oeuvre II, Paris, pp. 317–328, 1826. 107Google Scholar
  5. 5.
    D. Grigor’ev, N. Vorobjov. Solving systems of polynomial inequalities in subexponential time. J. Symbolic Comput., Vol. 5, pp. 37–64, 1988. 100, 107MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Haudorff. Set Theory. Chelsea, New York, 1962. 100, 101Google Scholar
  7. 7.
    J. G. Hocking and G. S. Young Topology. Dover Publications Inc., New York, 1988. 100MATHGoogle Scholar
  8. 8.
    H. W. Kuhn. Solvability and consistency for linear equations and inequalities. Amer. Math. Monthly, Vol. 63, pp 217–232, 1956. 107CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    L. Hörmander. The analysis of linear partial differential operators. Springer-Verlag Berlin, Vol. 2, 1983. 99Google Scholar
  10. 10.
    Y. Matiiassevitch. Enumerable sets are diophantine. Doklady Akad.Nauk SSSR, Vol. 191, pp 279–282, 1970 (English translation: Soviet Math. Doklady, pp 354–357, 1970). 106, 107Google Scholar
  11. 11.
    T. S. Motzkin. Beitrage zur theorie der linearen ungleichungen. Azriel: Jerusalem, 1936. 107Google Scholar
  12. 12.
    C. Ronse and M. Tajine. Discretization in Hausdorff Space. Journal of Mathematical Imaging & Vision, Vol. 12, no 3, pp. 219–242, 2000. 99, 100, 101MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. Ronse and M. Tajine. Hausdorff discretization for cellular distances, and its relation to cover and supercover discretization. To be revised, 2000. 99, 101Google Scholar
  14. 14.
    A. Seidenberg. A new decision method for elementary algebra. Ann. of Math., Vol. 60, pp 365–374, 1954. 99, 105CrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Tajine and C. Ronse. Preservation of topology by Hausdorff discretization and comparison to other discretization schemes. Submitted, 1999. 99, 100, 101, 103Google Scholar
  16. 16.
    M. Tajine and C. Ronse. Hausdorff sampling of closed sets in a boundedly compact space. In preparation, 2000. 99Google Scholar
  17. 17.
    M. Tajine and C. Ronse. Topological Properties of Hausdorff discretizations. International Symposium on Mathematical Morphology 2000 (ISMM’2000), Palo Alto CA, USA. Kluwer Academic Publishers, pp. 41–50, 2000. 99Google Scholar
  18. 18.
    M. Tajine, D. Wagner and C. Ronse. Hausdorff discretization and its comparison with other discretization schemes. DGCI’99, Paris,LNCS Springer-Verlag, Vol. 1568, pp. 399–410, 1999. 99Google Scholar
  19. 19.
    A. Tarski. Sur les ensembles définissables de nombres réels. Fund. Math., Vol. 17, pp. 210–239, 1931. 99, 105MATHGoogle Scholar
  20. 20.
    A. Tarski. A decision method for elementary algebra and geometry. Tech. Rep., University of California Press, Berkeley and Los Angeles, 1951. 99, 105MATHGoogle Scholar
  21. 21.
    D. Wagner. Distance de Hausdorff et probléme discret-continu. Mémoire de D. E. A. (M.Sc. Dissertation), Université Louis Pasteur, Strasbourg (France). 1997. 99, 100, 101Google Scholar
  22. 22.
    D. Wagner, M. Tajine and C. Ronse. An approach to discretization based on the Hausdorff metric.ISMM’1998. Kluwer Academic Publishers. pp. 67–74, 1998. 99, 100, 101Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Mohamed Tajine
    • 1
  • Christian Ronse
    • 1
  1. 1.Laboratoire des Sciences de l’Imagede l’Informatique et de la Télédétection (LSIIT, UPRES-A CNRS 7005)IllkirchFrance

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