×

Unimodular covers of multiples of polytopes. (English) Zbl 1044.52006

Let \(P\subset \mathbb R^n\) be an \(n\)-dimensional lattice polytope, i.e., a polytope whose vertices belong to the standard lattice. G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat [Toroidal embeddings. I. (1973; Zbl 0271.14017)] have shown that there exists a constant \(c_P\) (depending on \(P\)) such that \(c_P\,P\) admits a regular triangulation into unimodular simplices. However, it is still an open (and challenging) problem to decide whether there exists a constant \(c_{ut}\) such that for all \(c \geq c_{ut}\) the polytope \(c\,P\) admits such a triangulation and whether there exists a uniform bound (independent of \(P\)) for such a \(c_{ut}\).
In the paper under review the authors study a closely related problem, namely they look for unimodular coverings instead of triangulations and obtain the following beautiful and remarkable result. There exists a constant \(c_{uc}(n)\) only depending on the dimension such that for all \(c\geq c_{uc}(n)\) the polytope \(c\,P\) can be covered by unimodular simplices. The constructive proof gives an upper bound on \(c_{uc}(n)\) of size \(O(n^5\,(3/2)^{n^{3/2}})\). A priori it was not clear at all that there really exists such a constant and it was only known that \(c_{uc}(2)=1\) and \(c_{uc}(3)=2\). They prove their result by passage to rational cones, for which they establish a similar bound.
It is also worth mentioning that only one argument in their proof leads to the exponential term in the bound. Namely, they give an exponential upper bound on the height of vectors which are needed for a unimodular triangulation of a simplical cone.

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H06 Lattices and convex bodies (number-theoretic aspects)

Citations:

Zbl 0271.14017
PDFBibTeX XMLCite
Full Text: arXiv EuDML EMIS