Euclideaness and final polynomials in oriented matroid theory

Combinatorica volume 13, pages259–268(1993)Cite this article

Abstract

This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programs in non-euclidean oriented matroids.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. [1]

    A. Björner. M. Las Vergnas, B. Sturmfels, N. White, andG. M. Ziegler:Oriented Matroids, Cambridge University Press, Encyclopedia of Mathematics and its Applications,46, 1993.

  2. [2]

    R. G. Bland: A combinatorial abstraction of linear programming,J. Combinatorial Theory B 23 (1977), 33–57.

    Google Scholar 

  3. [3]

    J. Bokowski, andJ. Richter: On the finding of final polynomials,Europ. J. Combinatorics 11 (1990), 21–34.

    Google Scholar 

  4. [4]

    J. Bokowski, andJ. Richter: On the classification of non-realizable oriented Matroids, Part II: Properties. Preprint, TH-Darmstadt (1990).

  5. [5]

    J. Bokowski, andB. Sturmfels:Computational Synthetic Geometry, Lecture Notes in Mathematics1355, Springer, Berlin, 1989.

    Google Scholar 

  6. [6]

    K. Fukuda:Oriented matroid programming, Ph.D. Thesis, University of Waterloo, 1982.

  7. [7]

    A. Mandel:Topology of Oriented Matroids, Ph.D. Thesis, University of Waterloo, 1982.

  8. [8]

    J. Richter-Gebert:On the realizability problem of combinatorial geometries — decision methods, Dissertation, TH-Darmstadt, 1991.

  9. [9]

    P. W. Shor: Stretchability of Pseudolines is NP-Hard, in:Applied Geometry and Discrete Mathematics — The Victor Klee Festschrift (P. Gritzmann, B. Sturmfels, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc., Providence, RI,4, 531–554.

Download references

Author information

Affiliations

  1. Auf dem Wingert 3, D-64380, Roßdorf, Germany

    Jürgen Richter-Gebert

Authors
  1. Jürgen Richter-Gebert
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Richter-Gebert, J. Euclideaness and final polynomials in oriented matroid theory. Combinatorica 13, 259–268 (1993). https://doi.org/10.1007/BF01202352

Download citation

AMS subject classification code (1991)

  • 05 B 35
  • 51 A 25