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Gröbner bases and triangulations of the second hypersimplex

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Abstract

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.

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Authors and Affiliations

  1. Center for Applied Mathematics, Cornell University, 14583, Ithaca, NY

    Jesús A. De Loera

  2. Department of Mathematics, Cornell University, 14583, Ithaca, NY

    Bernd Sturmfels

  3. School of Operations Research and Industrial Engineering, Cornell University, 14583, Ithaca, NY

    Rekha R. Thomas

Authors
  1. Jesús A. De Loera
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  2. Bernd Sturmfels
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  3. Rekha R. Thomas
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Additional information

Research partially supported by a doctoral fellowship of the National University of Mexico, the National Science Foundation, the David and Lucile Packard Foundation and the U.S. Army Research Office (through ACSyAM/MSI, Cornell).

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De Loera, J.A., Sturmfels, B. & Thomas, R.R. Gröbner bases and triangulations of the second hypersimplex. Combinatorica 15, 409–424 (1995). https://doi.org/10.1007/BF01299745

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