Publications (9)4.35 Total impact
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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 BrionVergne 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 BrionVergne 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 finitedimensional 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 ArdilaPostnikov and HoltzRon in the context of zonotopal algebra and power ideals.  [Show abstract] [Hide abstract]
ABSTRACT: We show that the fvector of the matroid complex of a representable matroid is logconcave. 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 realvalued 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 socalled 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 KhovanskiiPukhlikov formula that relates the volume and the number of integer points in a smooth lattice polytope.  [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 realvalued 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 socalled 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 deletioncontraction decomposition.  [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 DahmenMicchelli 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 ArdilaPostnikov, HoltzRon, HoltzRonXu, LiRon, 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. 
Article: Matroids and logconcavity
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ABSTRACT: We show that fvectors of matroid complexes of realisable matroids are logconcave. 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 logconcave sequence. We also discuss the relationship between logconcavity of fvectors and hvectors of matroids. In the last section we explain the connection between zonotopal algebra and fvectors and characteristic polynomials of matroids.  [Show abstract] [Hide abstract]
ABSTRACT: We show that fvectors of matroid complexes of realizable matroids are strictly logconcave. 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 logconcave sequence. We also prove a statement on logconcavity of hvectors 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 logconcavity results and various matroid/graph polynomials. 
Article: Hierarchical Zonotopal Power Ideals
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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 ArdilaPostnikov on power ideals and by HoltzRon and HoltzRonXu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules that were introduced by SturmfelsXu.  [Show abstract] [Hide abstract]
ABSTRACT: NOTE: Unfortunately, most of the results mentioned here were already known under the name of "dseparated 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 2dimensional case by Tardos and for the general case by Kaiser [8]. An excellent survey article ("Transversals of dintervals') is available on http://www.renyi.hu/~tardos. Still, we leave this paper available to the public on http://page.mi.fuberlin.de/dawerner, also because one might find the references useful.  We study the following Gallaitype 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 axisparallel hyperplane intersecting at least two of the objects. What can we say about the number of axisparallel 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 superexponential 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
Publication Stats
38  Citations  
4.35  Total Impact Points  
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Institutions

2013

University of Oxford
Oxford, England, United Kingdom


2010

Technische Universität Berlin
Berlín, Berlin, Germany
