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Transactions of the Moscow Mathematical Society

This journal, a translation of Trudy Moskovskogo Matematicheskogo Obshchestva, contains the results of original research in pure mathematics.

ISSN 1547-738X (online) ISSN 0077-1554 (print)

The 2020 MCQ for Transactions of the Moscow Mathematical Society is 0.74.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Unimodular triangulations of dilated 3-polytopes
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by F. Santos and G. M. Ziegler
Trans. Moscow Math. Soc. 2013, 293-311
Published electronically: April 9, 2014


A seminal result in the theory of toric varieties, by Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope $P$ there is a positive integer $k$ such that the dilated polytope $kP$ has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that $k=4$ works for every polytope. But this does not imply that every $k>4$ works as well. We here study the values of $k$ for which the result holds, showing that:

  1. It contains all composite numbers.

  2. It is an additive semigroup.

These two properties imply that the only values of $k$ that may not work (besides $1$ and $2$, which are known not to work) are $k\in \{3,5,7,11\}$. With an ad-hoc construction we show that $k=7$ and $k=11$ also work, except in this case the triangulation cannot be guaranteed to be “standard” in the boundary. All in all, the only open cases are $k=3$ and $k=5$.

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Bibliographic Information
  • F. Santos
  • Affiliation: Facultad de Ciencias, Universidad de Cantabria, Spain
  • MR Author ID: 360182
  • ORCID: 0000-0003-2120-9068
  • Email:
  • G. M. Ziegler
  • Affiliation: Inst. Mathematics, FU Berlin, Germany
  • Email:
  • Published electronically: April 9, 2014
  • Additional Notes: The work of the first author was supported in part by the Spanish Ministry of Science under Grants MTM2011-22792 and by MICINN-ESF EUROCORES programme EuroGIGA— ComPoSe — IP04 (Project EUI-EURC-2011-4306). Part of this work was done while the first author was visiting FU Berlin in 2012 and 2013 supported by a Research Fellowship of the Alexander von Humboldt Foundation.
    The work of the second author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 247029-SDModels and by the DFG Research Center Matheon “Mathematics for Key Technologies” in Berlin
  • © Copyright 2014 F. Santos, G. M. Ziegler
  • Journal: Trans. Moscow Math. Soc. 2013, 293-311
  • MSC (2010): Primary 52B20, 14M25
  • DOI:
  • MathSciNet review: 3235802