h*-Polynomials with Roots on the Unit Circle

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For an n-dimensional lattice simplex (Formula presented.) with vertices given by the standard basis vectors and (Formula presented.) where (Formula presented.) has positive entries, we investigate when the Ehrhart (Formula presented.) -polynomial for (Formula presented.) factors as a product of geometric series in powers of z. Our motivation is a theorem of Rodriguez-Villegas implying that when the (Formula presented.) -polynomial of a lattice polytope P has all roots on the unit circle, then the Ehrhart polynomial of P has positive coefficients. We focus on those (Formula presented.) for which (Formula presented.) has only two or three distinct entries, providing both theoretical results and conjectures/questions motivated by experimental evidence.

Original languageEnglish
Pages (from-to)332-348
Number of pages17
JournalExperimental Mathematics
Issue number3
StatePublished - 2021

Bibliographical note

Funding Information:
The first author was partially supported by grant H98230-16-1-0045 from the U.S. National Security Agency. The second author was partially supported by a grant from the Simons Foundation #426756. This material is also based in part upon work supported by the National Science Foundation under Grant No. DMS-1440140 while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.

Publisher Copyright:
© 2019 Taylor & Francis Group, LLC.


  • Ehrhart positivity
  • h*-polynomial
  • lattice simplex
  • unit circle rooted

ASJC Scopus subject areas

  • Mathematics (all)


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