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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

   

 

Unimodular triangulations of dilated 3-polytopes


Authors: F. Santos and G. M. Ziegler
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 74 (2013), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2013, 293-311
MSC (2010): Primary 52B20, 14M25
Published electronically: April 9, 2014
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Abstract: 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: 2.3

($ 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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Additional Information

F. Santos
Affiliation: Facultad de Ciencias, Universidad de Cantabria, Spain
Email: francisco.santos@unican.es

G. M. Ziegler
Affiliation: Inst. Mathematics, FU Berlin, Germany
Email: ziegler@math.fu-berlin.de

DOI: http://dx.doi.org/10.1090/S0077-1554-2014-00220-X
PII: S 0077-1554(2014)00220-X
Keywords: lattice polytopes, unimodular triangulations, KKMS theorem
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
Article copyright: © Copyright 2014 F. Santos, G.M. Ziegler