Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Thorsten Theobald
Goethe-Universität Frankfurt am Main
Combinatorial Aspects of Tropical Intersections and Self-Intersections
In this talk we consider two foundational combinatorial questions on the intersection of tropical hypersurfaces.
Given tropical polynomials g1,...,gk in n variables with Newton polytopes P1,...,Pk, we first study
the f -vector, the number of unbounded faces and (in case of a curve) the genus. For the case of curves,
the second question is concerned with the (unweighted) number of self-intersection points under linear
projections onto the plane.
Along studying these questions, we meet some intriguing connections between certain mixed volumes
and alternating sums of integer points in Minkowski sums of polytopes, a mixed version of Ehrhart theory,
as well as mixed fiber polytopes. While for the first problem, our characterizations are exact, for the second
question our main results are bounds as well as constructions with many self-intersections.
(Partially based on joint work with Kerstin Hept and with Reinhard Steffens.)
Christian Haase
Goethe-Universität Frankfurt am Main
Computing Toric Ideals of Integrally Closed Polytopes
A lattice polytope is the convex hull of finitely many points in the integer lattice. The toric ideal of such a
polytope P is an ideal in a polynomial ring which encodes affine dependencies among the lattice points in
P.
Generating sets for these toric ideals have been used in integer programming, algebraic statistics, and
many other mathematical fields. In this talk, I want to present a novel approach to compute the toric ideal
of P and report on experiments with a first implementation of the algorithm.
It works best under the (checkable) additional assumption that P be integrally closed. That is, we want
that every lattice point in a dilation kP of P can be written as the sum of k lattice points in P.
An extension of the algorithm has the potential to compute higher Betti numbers of toric ideals.
(This is joint work in progress with Benjamin Lorenz.)
Matthias Henze
Otto-von-Guericke-Universität Magdeburg
Blichfeldt-Type Inequalities and Central Symmetry
A classical result of Blichfeldt, which dates back to 1921, gives a sharp lower bound on the volume
of a convex body K whose lattice points span the whole space in terms of the lattice point enumerator
#(K ∩Z n ). We are interested in a version of this inequality on the set of centrally symmetric convex bodies.
Our motivation to study this problem comes from a lack of methods that exploit the symmetry assumption
in problems of a similar kind and where central symmetry is a natural condition.
We report upon first results for special families of centrally symmetric convex bodies.
(This is joint ongoing work with Martin Henk.)
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