[Show abstract][Hide abstract] ABSTRACT: Let $X$ be a $(d\times N)$-matrix. We consider the variable polytope
$\Pi_X(u) = \{ w \ge 0 : X w = u \}$. It is known that the function $T_X$ that
assigns to a parameter $u \in \mathbb{R}^d$ the volume of the polytope
$\Pi_X(u)$ is piecewise polynomial. The Brion-Vergne formula implies that the
number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain
differential operator to the function $T_X$.
In this article we slightly improve the Brion-Vergne formula and we study two
spaces of differential operators that arise in this context: the space of
relevant differential operators (i.e. operators that do not annihilate $T_X$)
and the space of nice differential operators (i.e. operators that leave $T_X$
continuous). These two spaces are finite-dimensional homogeneous vector spaces
and their Hilbert series are evaluations of the Tutte polynomial of the
arithmetic matroid defined by the matrix $X$. They are closely related to the
$\mathcal{P}$-spaces studied by Ardila-Postnikov and Holtz-Ron in the context
of zonotopal algebra and power ideals.
Journal of Algebraic Combinatorics 08/2014; DOI:10.1007/s10801-015-0621-2 · 0.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that the f-vector of the matroid complex of a representable matroid is log-concave. This proves the representable case of a conjecture made by Mason in 1972.
[Show abstract][Hide abstract] ABSTRACT: Let $X$ be a list of vectors that is totally unimodular. In a previous
article the author proved that every real-valued function on the set of
interior lattice points of the zonotope defined by $X$ can be extended to a
function on the whole zonotope of the form $p(D)B_X$ in a unique way, where
$p(D)$ is a differential operator that is contained in the so-called internal
$\Pcal$-space. In this paper we construct an explicit solution to this
interpolation problem in terms of Todd operators. As a corollary we obtain a
slight generalisation of the Khovanskii-Pukhlikov formula that relates the
volume and the number of integer points in a smooth lattice polytope.
International Mathematics Research Notices 05/2013; 2015(14). DOI:10.1093/imrn/rnu095 · 1.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let $X$ be a totally unimodular list of vectors in some lattice. Let $B_X$ be
the box spline defined by $X$. Its support is the zonotope $Z(X)$. We show that
any real-valued function defined on the set of lattice points in the interior
of $Z(X)$ can be extended to a function on $Z(X)$ of the form $p(D)B_X$ in a
unique way, where $p(D)$ is a differential operator that is contained in the
so-called internal $\Pcal$-space. This was conjectured by Olga Holtz and Amos
Ron. We also point out connections between this interpolation problem and
matroid theory, including a deletion-contraction decomposition.
International Mathematics Research Notices 11/2012; DOI:10.1093/imrn/rnt142 · 1.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Zonotopal algebra is the study of a family of pairs of dual vector spaces of
multivariate polynomials that can be associated with a list of vectors X. It
connects objects from combinatorics, geometry, and approximation theory. The
origin of zonotopal algebra is the pair (D(X),P(X)), where D(X) denotes the
Dahmen-Micchelli space that is spanned by the local pieces of the box spline
and P(X) is a space spanned by products of linear forms.
The first main result of this paper is the construction of a canonical basis
for D(X). We show that it is dual to the canonical basis for P(X) that is
already known.
The second main result of this paper is the construction of a new family of
zonotopal spaces that is far more general than the ones that were recently
studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We
call the underlying combinatorial structure of those spaces forward exchange
matroid. A forward exchange matroid is an ordered matroid together with a
subset of its set of bases that satisfies a weak version of the basis exchange
axiom.
[Show abstract][Hide abstract] ABSTRACT: We show that f-vectors of matroid complexes of realisable matroids are
log-concave. This was conjectured by Mason in 1972. Our proof uses the recent
result by Huh and Katz who showed that the coefficients of the characteristic
polynomial of a realisable matroid form a log-concave sequence. We also discuss
the relationship between log-concavity of f-vectors and h-vectors of matroids.
In the last section we explain the connection between zonotopal algebra and
f-vectors and characteristic polynomials of matroids.
[Show abstract][Hide abstract] ABSTRACT: We show that f-vectors of matroid complexes of realizable matroids are strictly log-concave. This was conjectured by Mason in 1972. Our proof uses the recent result by Huh and Katz who showed that the coefficients of the characteristic polynomial of a realizable matroid form a log-concave sequence. We also prove a statement on log-concavity of h-vectors which strengthens a result by Brown and Colbourn. In the last two sections, we give a brief introduction to zonotopal algebra and we explain how it relates to our log-concavity results and various matroid/graph polynomials.
[Show abstract][Hide abstract] ABSTRACT: Zonotopal algebra deals with ideals and vector spaces of polynomials that are
related to several combinatorial and geometric structures defined by a finite
sequence of vectors. Given such a sequence X, an integer k>=-1 and an upper set
in the lattice of flats of the matroid defined by X, we define and study the
associated hierarchical zonotopal power ideal. This ideal is generated by
powers of linear forms. Its Hilbert series depends only on the matroid
structure of X. Via the Tutte polynomial, it is related to various other
matroid invariants, e.g. the shelling polynomial and the characteristic
polynomial.
This work unifies and generalizes results by Ardila-Postnikov on power ideals
and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also
generalize a result on zonotopal Cox modules that were introduced by
Sturmfels-Xu.
European Journal of Combinatorics 11/2010; 33(6). DOI:10.1016/j.ejc.2012.01.004 · 0.65 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: NOTE: Unfortunately, most of the results mentioned here were already known under the name of "d-separated interval piercing". The result that T_d(m) exists was first proved by Gya\'rfa\'s and Lehel in 1970, see [5]. Later, the result was strengthened by Ka\'rolyi and Tardos [9] to match our result. Moreover, their proof (in a different notation, of course) uses ideas very similar to ours and leads to a similar recurrence. Also, our conjecture turns out to be right and was proved for the 2-dimensional case by Tardos and for the general case by Kaiser [8]. An excellent survey article ("Transversals of d-intervals') is available on http://www.renyi.hu/~tardos. Still, we leave this paper available to the public on http://page.mi.fu-berlin.de/dawerner, also because one might find the references useful. ----- We study the following Gallai-type of problem: Assume that we are given a family X of convex objects in R^d such that among any subset of size m, there is an axis-parallel hyperplane intersecting at least two of the objects. What can we say about the number of axis-parallel hyperplanes that sufficient to intersect all sets in the family? In this paper, we show that this number T_d(m) exists, i.e., depends only on m and the dimension d, but not on the size of the set X. First, we derive a very weak super-exponential bound. Using this result, by a simple proof we are able to show that this number is even polynomially bounded for any fixed d. We partly answer open problem 74 on http://maven.smith.edu/~orourke/TOPP/, where the planar case is considered, by improving the best known exponential bound to O(m^2). Comment: Withdrawn, because we found out that most of the results were already known (under a different name). See updated abstract for details