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Irrational Triangles with Polynomial Ehrhart Functions

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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.

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Notes

  1. This might seem like an abuse of notation, since we also defined a function \(\sigma (t)\) in (2.4). Notice, however, that under the assumptions of Proposition 2.2, in particular under the assumption that \(\alpha \) and \(\alpha \beta \) are both integers, the functions in (2.4) and (2.9) agree.

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA, 95060, USA

    Dan Cristofaro-Gardiner

  2. School of Mathematics and Statistics, Southwest University, Chongqing, People’s Republic of China

    Teresa Xueshan Li

  3. Department of Mathematics, University of Miami, Coral Gables, FL, 33124, USA

    Richard P. Stanley

Authors
  1. Dan Cristofaro-Gardiner
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  2. Teresa Xueshan Li
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  3. Richard P. Stanley
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Correspondence to Richard P. Stanley.

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The first author would like to thank Martin Gardiner for helpful discussions. The first author was partially supported by NSF Grants DMS-1402200 and DMS-1711976. The second author was supported by the National Science Foundation of China (Grant No. 11601440), the Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0245) and Fundamental Research Funds for Central Universities (Grant No. XDJK2018C075). The third author was partially supported by NSF Grant DMS-1068625.

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Cristofaro-Gardiner, D., Li, T.X. & Stanley, R.P. Irrational Triangles with Polynomial Ehrhart Functions. Discrete Comput Geom 61, 227–246 (2019). https://doi.org/10.1007/s00454-018-0036-7

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  • DOI: https://doi.org/10.1007/s00454-018-0036-7

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