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.
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E. Andres. Standard Cover: a new class of discrete primitives. Internal Report, IRCOM, Université de Poitiers. 100, 107
J. Bochnak, M. Coste, M.-F. Roy. Real Algebraic Geometry. Serie of Modern Surveys in Mathematics, Vol. 36, Springer, 1998. 105
H. Busemann The geometry of geodesics. Academec Press, New York, 1955. 100, 101
J. B. J. Fourier. Solution d’une question particuliére du calcul des inégalités. Oeuvre II, Paris, pp. 317–328, 1826. 107
D. Grigor’ev, N. Vorobjov. Solving systems of polynomial inequalities in subexponential time. J. Symbolic Comput., Vol. 5, pp. 37–64, 1988. 100, 107
F. Haudorff. Set Theory. Chelsea, New York, 1962. 100, 101
J. G. Hocking and G. S. Young Topology. Dover Publications Inc., New York, 1988. 100
H. W. Kuhn. Solvability and consistency for linear equations and inequalities. Amer. Math. Monthly, Vol. 63, pp 217–232, 1956. 107
L. Hörmander. The analysis of linear partial differential operators. Springer-Verlag Berlin, Vol. 2, 1983. 99
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, 107
T. S. Motzkin. Beitrage zur theorie der linearen ungleichungen. Azriel: Jerusalem, 1936. 107
C. Ronse and M. Tajine. Discretization in Hausdorff Space. Journal of Mathematical Imaging & Vision, Vol. 12, no 3, pp. 219–242, 2000. 99, 100, 101
C. Ronse and M. Tajine. Hausdorff discretization for cellular distances, and its relation to cover and supercover discretization. To be revised, 2000. 99, 101
A. Seidenberg. A new decision method for elementary algebra. Ann. of Math., Vol. 60, pp 365–374, 1954. 99, 105
M. Tajine and C. Ronse. Preservation of topology by Hausdorff discretization and comparison to other discretization schemes. Submitted, 1999. 99, 100, 101, 103
M. Tajine and C. Ronse. Hausdorff sampling of closed sets in a boundedly compact space. In preparation, 2000. 99
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. 99
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. 99
A. Tarski. Sur les ensembles définissables de nombres réels. Fund. Math., Vol. 17, pp. 210–239, 1931. 99, 105
A. Tarski. A decision method for elementary algebra and geometry. Tech. Rep., University of California Press, Berkeley and Los Angeles, 1951. 99, 105
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, 101
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, 101
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Tajine, M., Ronse, C. (2000). Hausdorff Discretizations of Algebraic Sets and Diophantine Sets. In: Borgefors, G., Nyström, I., di Baja, G.S. (eds) Discrete Geometry for Computer Imagery. DGCI 2000. Lecture Notes in Computer Science, vol 1953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44438-6_9
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DOI: https://doi.org/10.1007/3-540-44438-6_9
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