1,791 research outputs found
Volume and lattice points of reflexive simplices
We prove sharp upper bounds on the volume and the number of lattice points on
edges of higher-dimensional reflexive simplices. These convex-geometric results
are derived from new number-theoretic bounds on the denominators of unit
fractions summing up to one. The main algebro-geometric application is a sharp
upper bound on the anticanonical degree of higher-dimensional Q-factorial
Gorenstein toric Fano varieties with Picard number one, where we completely
characterize the case of equality.Comment: AMS-LaTeX, 19 pages; paper reorganized, introduction added,
bibliography updated; typos correcte
Lattice Delone simplices with super-exponential volume
In this short note we give a construction of an infinite series of Delone
simplices whose relative volume grows super-exponentially with their dimension.
This dramatically improves the previous best lower bound, which was linear.Comment: 7 pages; v2: revised version improves our exponential lower bound to
a super-exponential on
Lower bounds for the simplexity of the n-cube
In this paper we prove a new asymptotic lower bound for the minimal number of
simplices in simplicial dissections of -dimensional cubes. In particular we
show that the number of simplices in dissections of -cubes without
additional vertices is at least .Comment: 10 page
The Hyperdeterminant and Triangulations of the 4-Cube
The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in
16 unknowns which has 2894276 terms. We compute the Newton polytope of this
polynomial and the secondary polytope of the 4-cube. The 87959448 regular
triangulations of the 4-cube are classified into 25448 D-equivalence classes,
one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest
regular subdivisions, one for each facet of the secondary polytope, but only
268 of them come from the hyperdeterminant.Comment: 30 pages, 6 figures; An author's name changed, typos fixe
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