The Mathematical Intelligencer

, Volume 31, Issue 1, pp 9–17 | Cite as

Open image in new window (Formulas of Brion, Lawrence, and Varchenko on rational generating functions for cones)



Edge Direction Mathematical Intelligencer Tangent Cone Integer Point Simple Cone 
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Research of Beck supported in part by NSF grant DMS-0810105. Research of Haase supported in part by NSF grant DMS-0200740 and a DFG Emmy Noether fellowship. Research of Sottile supported in part by the Clay Mathematical Institute and NSF CAREER grant DMS-0538734.


  1. 1.
    A.I. Barvinok, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res. 19 (1994), 769–779.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A.I. Barvinok, A course in convexity, Graduate Studies in Mathematics, vol. 54, American Mathematical Society, Providence, RI, 2002.Google Scholar
  3. 3.
    P. Baum, Wm. Fulton, and G. Quart, Lefschetz-Riemann-Roch for singular varieties. Acta Math. 143 (1979), no. 3–4, 193–211.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Beck and S. Robins, Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, New York, 2007.Google Scholar
  5. 5.
    M. Beck and F. Sottile, Irrational proofs of three theorems of Stanley, 2005, European J. Combin. 28 (2007), 403–409.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C.J. Brianchon, Théorème nouveau sur les polyèdres, J. école (Royale) Polytechnique 15 (1837), 317–319.Google Scholar
  7. 7.
    M. Brion, Points entiers dans les polyèdres convexes, Ann. Sci. École Norm. Sup. 21 (1988), no. 4, 653–663.MATHMathSciNetGoogle Scholar
  8. 8.
    J.A. De Loera, D. Haws, R. Hemmecke, P. Huggins, and R. Yoshida, A user’s guide for LattE v1.1, software package LattE (2004), electronically available at
  9. 9.
    J.P. Gram, Om rumvinklerne i et polyeder, Tidsskrift for Math. (Copenhagen) 4 (1874), no. 3, 161–163.Google Scholar
  10. 10.
    H. Groemer, On the extension of additive functionals on classes of convex sets, Pacific J. Math. 75 (1978), no. 2, 397–410.MATHMathSciNetGoogle Scholar
  11. 11.
    M.-N. Ishida, Polyhedral Laurent series and Brion’s equalities, Internat. J. Math. 1 (1990), no. 3, 251–265.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A.G. Khovanskii and A.V. Pukhlikov, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz 4 (1992), 188–216.MathSciNetGoogle Scholar
  13. 13.
    M. Koeppe, A primal Barvinok algorithm based on irrational decompositions, SIAM J. Discrete Math. 21 (2007), no. 1, 220–236.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    J. Lawrence, Valuations and polarity, Discrete Comput. Geom. 3 (1988), no. 4, 307–324.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. Lawrence, Polytope volume computation, Math. Comp. 57 (1991), no. 195, 259–271.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A.N. Varchenko, Combinatorics and topology of the arrangement of affine hyperplanes in the real space, Funktsional. Anal. i Prilozhen. 21 (1987), no. 1, 11–22.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Verdoolaege, software package barvinok (2004), electronically available at
  18. 18.
    W. Volland, Ein Fortsetzungssatz für additive Eipolyhederfunk- tionale im euklidischen Raum, Arch. Math. 8 (1957), 144– 149.MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Fachbereich Mathematik & InformatikFreie Universität BerlinBerlinGermany
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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