Skip to main content
Log in

(Formulas of Brion, Lawrence, and Varchenko on rational generating functions for cones)

  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  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.

    Article  MATH  MathSciNet  Google Scholar 

  2. A.I. Barvinok, A course in convexity, Graduate Studies in Mathematics, vol. 54, American Mathematical Society, Providence, RI, 2002.

  3. P. Baum, Wm. Fulton, and G. Quart, Lefschetz-Riemann-Roch for singular varieties. Acta Math. 143 (1979), no. 3–4, 193–211.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Beck and S. Robins, Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, New York, 2007.

  5. M. Beck and F. Sottile, Irrational proofs of three theorems of Stanley, 2005, European J. Combin. 28 (2007), 403–409.

    Article  MATH  MathSciNet  Google Scholar 

  6. C.J. Brianchon, Théorème nouveau sur les polyèdres, J. école (Royale) Polytechnique 15 (1837), 317–319.

    Google Scholar 

  7. M. Brion, Points entiers dans les polyèdres convexes, Ann. Sci. École Norm. Sup. 21 (1988), no. 4, 653–663.

    MATH  MathSciNet  Google Scholar 

  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 http://www.math.ucdavis.edu/~latte/.

  9. J.P. Gram, Om rumvinklerne i et polyeder, Tidsskrift for Math. (Copenhagen) 4 (1874), no. 3, 161–163.

    Google Scholar 

  10. H. Groemer, On the extension of additive functionals on classes of convex sets, Pacific J. Math. 75 (1978), no. 2, 397–410.

    MATH  MathSciNet  Google Scholar 

  11. M.-N. Ishida, Polyhedral Laurent series and Brion’s equalities, Internat. J. Math. 1 (1990), no. 3, 251–265.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  13. M. Koeppe, A primal Barvinok algorithm based on irrational decompositions, SIAM J. Discrete Math. 21 (2007), no. 1, 220–236.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Lawrence, Valuations and polarity, Discrete Comput. Geom. 3 (1988), no. 4, 307–324.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Lawrence, Polytope volume computation, Math. Comp. 57 (1991), no. 195, 259–271.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Verdoolaege, software package barvinok (2004), electronically available at http://freshmeat.net/projects/barvinok/.

  18. W. Volland, Ein Fortsetzungssatz für additive Eipolyhederfunk- tionale im euklidischen Raum, Arch. Math. 8 (1957), 144– 149.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matthias Beck.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beck, M., Haase, C. & Sottile, F. (Formulas of Brion, Lawrence, and Varchenko on rational generating functions for cones). Math Intelligencer 31, 9–17 (2009). https://doi.org/10.1007/s00283-008-9013-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00283-008-9013-y

Keywords

Navigation