Sabitov polynomials for volumes of polyhedra in four dimensions

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Abstract

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.

Original languageEnglish
Pages (from-to)586-611
Number of pages26
JournalAdvances in Mathematics
Volume252
DOIs
Publication statusPublished - 15 Feb 2014

Keywords

  • Cayley-Menger determinant
  • Flexible polyhedron
  • Place
  • Resultant
  • Simplicial complex
  • Volume

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