259 research outputs found
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
Stanley's Major Contributions to Ehrhart Theory
This expository paper features a few highlights of Richard Stanley's
extensive work in Ehrhart theory, the study of integer-point enumeration in
rational polyhedra. We include results from the recent literature building on
Stanley's work, as well as several open problems.Comment: 9 pages; to appear in the 70th-birthday volume honoring Richard
Stanle
A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope
We present a multivariate generating function for all n x n nonnegative
integral matrices with all row and column sums equal to a positive integer t,
the so called semi-magic squares. As a consequence we obtain formulas for all
coefficients of the Ehrhart polynomial of the polytope B_n of n x n
doubly-stochastic matrices, also known as the Birkhoff polytope. In particular
we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric
h-vectors of Gorenstein polytopes
We show that the Ehrhart h-vector of an integer Gorenstein polytope with a
regular unimodular triangulation satisfies McMullen's g-theorem; in particular,
it is unimodal. This result generalizes a recent theorem of Athanasiadis
(conjectured by Stanley) for compressed polytopes. It is derived from a more
general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a
field) by a "long" regular sequence in such a way that the quotient is still a
normal affine monoid algebra. This technique reduces all questions about the
Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with
exactly one interior lattice point, provided each lattice point in a multiple
cP, c in N, can be written as the sum of n lattice points in P. (Up to a
translation, the polytope Q belongs to the class of reflexive polytopes
considered in connection with mirror symmetry.) If P has a regular unimodular
triangulation, then it follows readily that the Ehrhart h-vector of P coincides
with the combinatorial h-vector of the boundary complex of a simplicial
polytope, and the g-theorem applies.Comment: 12 pages; besides minor modifications the main result needs the
additional assumption that the polytope P has a regular unimodular
triangulation. The extra hypothesis has been included as well as the crucial
construction where it is use
Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices
In Ehrhart theory, the -vector of a rational polytope often provide
insights into properties of the polytope that may be otherwise obscured. As an
example, the Birkhoff polytope, also known as the polytope of real
doubly-stochastic matrices, has a unimodal -vector, but when even small
modifications are made to the polytope, the same property can be very difficult
to prove. In this paper, we examine the -vectors of a class of polytopes
containing real doubly-stochastic symmetric matrices.Comment: 11 pages; this revision removes an erroneous proposition from earlier
versions and expands on the implication
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