Publications (5)1.95 Total impact
 [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
[Show abstract] [Hide abstract] 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
[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 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
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35  Citations  
1.95  Total Impact Points  
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2010

Technische Universität Berlin
Berlín, Berlin, Germany
