On Positivity of Ehrhart Polynomials

Part of the Association for Women in Mathematics Series book series (AWMS, volume 16)


Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this chapter is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive and pose a few relevant questions.



The author is partially supported by a grant from the Simons Foundation #426756. The writing was completed when the author was attending the program “Geometric and Topological Combinatorics” at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester, and she was partially supported by the NSF grant DMS-1440140.

The author would like to thank Gabriele Balletti, Ben Braun, Federico Castillo, Ron King, Akihiro Higashitani, Sam Hopkins, Thomas McConville, Karola Mészáros, Alejandro Morales, Benjamin Nill, Andreas Paffenholz, Alex Postnikov, Richard Stanley, Liam Solus, and Akiyoshi Tsuchiya for valuable discussions and helpful suggestions. The author is particularly grateful to Federico Castillo and Alejandro Morales for their help in putting together some figures and data used in this chapter. Finally, the author is thankful to the two anonymous referees for their careful reading of this chapter and various insightful comments and suggestions.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA

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