, Volume 15, Issue 3, pp 409–424 | Cite as

Gröbner bases and triangulations of the second hypersimplex

  • Jesús A. De Loera
  • Bernd Sturmfels
  • Rekha R. Thomas


The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex Δ(2,n). We present a quadratic Gröbner basis for the associated toric idealK(K n ). The simplices in the resulting triangulation of Δ(2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK n . Forn≥6 the number of distinct initial ideals ofI(K n ) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(K n ) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Gröbner basis ofI(K n ) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.

Mathematics Subject Classification (1991)

52 B 12 13 P 10 05 C 50 


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Copyright information

© Akadémiai Kiadó 1995

Authors and Affiliations

  • Jesús A. De Loera
    • 1
  • Bernd Sturmfels
    • 2
  • Rekha R. Thomas
    • 3
  1. 1.Center for Applied MathematicsCornell UniversityIthaca
  2. 2.Department of MathematicsCornell UniversityIthaca
  3. 3.School of Operations Research and Industrial EngineeringCornell UniversityIthaca

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