Reciprocal domains and Cohen-Macaulay -complexes in .
Riemann sums over polytopes
It is well-known that the -th Riemann sum of a compactly supported function on the real line converges to the Riemann integral at a much faster rate than the standard rate of convergence if the sum is over the lattice, . In this paper we prove an n-dimensional version of this result for Riemann sums over polytopes.
Rigidity and flexibility of virtual polytopes
All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.
Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines
Étant donnés un système de racines d’une des familles A, B, C, D, F, G et le groupe abélien libre qu’il engendre, on calcule explicitement la série de croissance de ce groupe relativement à Les résultats s’interprètent en termes du polynôme d’Ehrhart de l’enveloppe convexe de .
Simple Vertices of Maximal Minor Polytopes
Six little squares and how their numbers grow.
Staircases in .
The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings
In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding . In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions...
The Ehrhart polynomial of a lattice -simplex.
The -vector of a Gorenstein toric ring of a compressed polytope.
The hypermetric cone on seven vertices.
The Newton polygon of a product of power series.
The number of vertices of a Fano polytope
Let be a Gorenstein, -factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.
Toric varieties in phylogenetics [Book]
Toric varieties, lattice points and Dedekind sums.
Tropical resultants for curves and stable intersection.
Typical sets and typical continuous functions
Unimodular covers of multiples of polytopes.
Universal counting of lattice points in polytopes.
Variétés horosphériques de Fano
Une variété horosphérique est une variété algébrique normale dans laquelle un groupe algébrique réductif opère avec une orbite ouverte fibrée en tores sur une variété de drapeaux. En particulier, les variétés toriques et les variétés de drapeaux sont horosphériques. Dans cet article, on classifie les variétés horosphériques de Fano en termes de certains polytopes rationnels qui généralisent les polytopes réflexifs considérés par V. Batyrev. Puis on obtient une majoration du degré des variétés horosphériques...