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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
Feynman graph polynomials
The integrand of any multi-loop integral is characterised after Feynman
parametrisation by two polynomials. In this review we summarise the properties
of these polynomials. Topics covered in this article include among others:
Spanning trees and spanning forests, the all-minors matrix-tree theorem,
recursion relations due to contraction and deletion of edges, Dodgson's
identity and matroids.Comment: 35 pages, references adde
Boolean Witt vectors and an integral Edrei-Thoma theorem
A subtraction-free definition of the big Witt vector construction was
recently given by the first author. This allows one to define the big Witt
vectors of any semiring. Here we give an explicit combinatorial description of
the big Witt vectors of the Boolean semiring. We do the same for two variants
of the big Witt vector construction: the Schur Witt vectors and the -typical
Witt vectors. We use this to give an explicit description of the Schur Witt
vectors of the natural numbers, which can be viewed as the classification of
totally positive power series with integral coefficients, first obtained by
Davydov. We also determine the cardinalities of some Witt vector algebras with
entries in various concrete arithmetic semirings.Comment: Final version. To appear in Selecta Mat
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