Discrete & Computational Geometry

, Volume 15, Issue 3, pp 253–264 | Cite as

Nonregular triangulations of products of simplices

  • J. A. de Loera
Article

Abstract

We exhibit a nonregular triangulation for the product of two tetrahedra. This answers a question by Gel'fand, Kapranov, and Zelevinsky. We also give a complete classification of the symmetry classes of regular triangulations of ▽2×▽3. Our nonregular triangulation of ▽3×▽3 can be extended to a nonregular triangulation of the six-dimensional cube. The four-dimensional cube is the smallest cube with a nonregular triangulation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Babson and L. Billera, The geometry of products of minors, Preprint, 1994.Google Scholar
  2. 2.
    L. Billera and B. Sturmfels, Fiber polytopes,Ann. of Math.,135 (1992), 527–549.CrossRefMathSciNetGoogle Scholar
  3. 3.
    J. A. de Loera, Triangulations of Polytopes and Computational Algebra, Ph.D. Thesis, Cornell University, 1995.Google Scholar
  4. 4.
    H. Edelsbrunner and N. R. Shah, Incremental topological flipping works for regular triangulations,Proc. 8th annual ACM Symposium on Computational Geometry, ACM Press, New York, 1992, pp. 43–52.Google Scholar
  5. 5.
    I. M. Gel'fand, M. M. Kapranov, and A. V. Zelevinsky, Hypergeometric functions and toric varieties,Functional Anal. Appl.,23(2) (1989), 12–26.MathSciNetGoogle Scholar
  6. 6.
    I. M. Gel'fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants of polynomials in several variables and triangulations of Newton polytopes.Algebra i Analiz,2 (1990), 1–62.Google Scholar
  7. 7.
    I. M. Gel'fand, M. M. Kapranov, and A. V. Zelevinsky,Multidimensional Determinants, Discriminants and Resultants, Birkhäuser, Boston, 1994.Google Scholar
  8. 8.
    M. Haiman, A simple and relatively efficient triangulation of then-cube,Discrete Comput. Geom.,6 (1991), 287–289.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    C. W. Lee, Regular triangulations of convex polytopes, inApplied Geometry and Discrete Mathematics—The Victor Klee Festschrift (P. Gritzmann and B. Sturmfels, eds.) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, 1991, pp. 443–456.Google Scholar
  10. 10.
    P. S. Mara, Triangulations for the cube,J. Combin. Theory Ser. A,20 (1976), 170–177.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. J. Risler, Construction d'hypersurfaces réelles (d'après Viro),Sém. N. Bourbaki (1992), 763.Google Scholar
  12. 12.
    M. E. Rudin, An unshellable triangulation of a tetrahedron.Bull. Amer. Math. Soc.,64 (1958), 90–91.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Stanley,Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983.MATHGoogle Scholar
  14. 14.
    B. Sturmfels, Gröbner bases of toric varieties,Tôhoku Math. J.,43(2) (1991), 249–261.MATHMathSciNetGoogle Scholar
  15. 15.
    B. Sturmfels, Sparse elimination theory, inComputational Algebraic Geometry and Commutative Algebra (D. Eisenbud and L. Robbiano, eds.) (Proceedings, Cortona, June 1991), Cambridge University Press, Cambridge, 1993, pp. 264–298.Google Scholar
  16. 16.
    G. Ziegler,Lectures on Polytopes, Springer-Verlag, New York, 1994.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • J. A. de Loera
    • 1
  1. 1.Center for Applied MathematicsCornell UniversityIthacaUSA

Personalised recommendations