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


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.


Ehrhart function Period collapse Irrational triangle P-recursive 

Mathematics Subject Classification




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


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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

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