6,184 research outputs found
Lattice-point generating functions for free sums of convex sets
Let \J and \K be convex sets in whose affine spans intersect at
a single rational point in \J \cap \K, and let \J \oplus \K = \conv(\J \cup
\K). We give formulas for the generating function {equation*} \sigma_{\cone(\J
\oplus \K)}(z_1,..., z_n, z_{n+1}) = \sum_{(m_1,..., m_n) \in t(\J \oplus \K)
\cap \Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points
in all integer dilates of \J \oplus \K in terms of \sigma_{\cone \J} and
\sigma_{\cone \K}, under various conditions on \J and \K. This work is
motivated by (and recovers) a product formula of B.\ Braun for the Ehrhart
series of \P \oplus \Q in the case where and \Q are lattice polytopes
containing the origin, one of which is reflexive. In particular, we find
necessary and sufficient conditions for Braun's formula and its multivariate
analogue.Comment: 17 pages, 2 figures, to appear in Journal of Combinatorial Theory
Series
A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes
The free sum is a basic geometric operation among convex polytopes. This note
focuses on the relationship between the normalized volume of the free sum and
that of the summands. In particular, we show that the normalized volume of the
free sum of full dimensional polytopes is precisely the product of the
normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra
Conferenc
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives
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
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