Constrained Triangulations, Volumes of Polytopes, and Unit Equations

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Given a polytope P in R d and a subset U of its vertices, is there a triangulation of P using d-simplices that all contain U? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of P. Our proof relates triangulations of P to triangulations of its “shadow”, a projection to a lower-dimensional space determined by U. In particular, we obtain a formula relating the volume of P with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations.

Original languageEnglish
Pages (from-to)69-99
Number of pages31
JournalPublicationes Mathematicae
Issue number1-2
Publication statusPublished - 2021


  • Constrained triangulations
  • Projections of polytopes
  • S-integers
  • Simplotopes
  • Unit equations
  • Volumes of polytopes

ASJC Scopus subject areas

  • Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing


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