Matthias Lenz

Technische Universität Berlin, Berlín, Berlin, Germany

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Publications (3)0 Total impact

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    Matthias Lenz
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    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.
    06/2011;
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    Matthias Lenz
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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 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.
    11/2010;
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    Daniel Werner, Matthias Lenz
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    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
    04/2010;

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3 Citations

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Institutions

  • 2010
    • Technische Universität Berlin
      Berlín, Berlin, Germany