Advertisement

Discrete & Computational Geometry

, Volume 61, Issue 2, pp 227–246 | Cite as

Irrational Triangles with Polynomial Ehrhart Functions

  • Dan Cristofaro-Gardiner
  • Teresa Xueshan Li
  • Richard P. StanleyEmail author
Article
  • 39 Downloads

Abstract

While much research has been done on the Ehrhart functions of integral and rational polytopes, little is known in the irrational case. In our main theorem, we determine exactly when the Ehrhart function of a right triangle with legs on the axes and slant edge with irrational slope is a polynomial. We also investigate several other situations where the period of the Ehrhart function of a polytope is less than the denominator of that polytope. For example, we give examples of irrational polytopes with polynomial Ehrhart function in any dimension, and we find triangles with periods dividing any even-index k-Fibonacci number, but with larger denominators.

Keywords

Ehrhart function Period collapse Irrational triangle P-recursive 

Mathematics Subject Classification

52B20 

Notes

Acknowledgements

We thank Bjorn Poonen for his help with Lemma 2.3. We also thank the anonymous referees for many extremely helpful comments and suggestions.

References

  1. 1.
    Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Beck, M., Robins, S., Sam, S.V.: Positivity theorems for solid-angle polynomials. Beitr. Algebra Geom. 51(2), 493–507 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Beck, M., Sottile, F.: Irrational proofs for three theorems of Stanley. Eur. J. Comb. 28(1), 403–409 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    De Loera, J.A., McAllister, T.B.: Vertices of Gelfand–Tsetlin polytopes. Discrete Comput. Geom. 32(4), 459–470 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ehrhart, E.: Sur les polyèdres rationnels homothétiques à \(n\) dimensions. C. R. Acad. Sci. Paris Ser. A 254, 616–618 (1962)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Haase, C., McAllister, T.B.: Quasi-period collapse and \({\rm GL}_n({\mathbb{Z}})\)-scissors congruence in rational polytopes. In: Beck, M. (ed.) Integer Points in Polyhedra. Contemporary Mathematics, vol. 452, pp. 115–122. American Mathematical Society, Providence, RI (2008)Google Scholar
  7. 7.
    McAllister, T.B., Woods, K.M.: The minimum period of the Ehrhart quasi-polynomial of a rational polytope. J. Comb. Theory Ser. A 109(2), 345–352 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    McDuff, D., Schlenk, F.: The embedding capacity of \(4\)-dimensional symplectic ellipsoids. Ann. Math. 175(3), 1191–1282 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Schlenk, F.: Symplectic embedding problems, old and new. Bull. Am. Math. Soc. 55(2), 139–182 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Stanley, R.P.: Enumerative Combinatorics, vol. II. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  11. 11.
    Woods, K.: Computing the period of an Ehrhart quasi-polynomial. Electron. J. Comb. 12, 1–12 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California Santa CruzSanta CruzUSA
  2. 2.School of Mathematics and StatisticsSouthwest UniversityChongqingPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of MiamiCoral GablesUSA

Personalised recommendations