Graduiertenkolleg: Methods for Discrete Structures

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Monday Lecture and Colloquium

Monday, December 10, 2007

Freie Universität Berlin
Institut für Informatik
Takustr. 9,
room 005

Lecture - 14:15

Martin Aigner - FU Berlin

A proof that is not yet in the Book



Colloquium - 16:00

Benjamin Nill - FU Berlin

The degree of lattice polytopes

Given a lattice polytope P, let G(n) be the number of lattice points in the n-th dilate of P. This function is a polynomial, called Ehrhart polynomial of P. Moreover, the generating function of the sequence of numbers G(n) is a rational function whose numerator is a polynomial with non-negative integers, which we call the h*-polynomial of P. These fundamental results are due to Ehrhart and Stanley. In this talk we deal with the degree of the h*-polynomial which we also call the degree of P. This invariant equals at most the dimension of the polytope, with equality if and only if P contains an interior lattice point. We propose that the degree may be regarded as a kind of "lattice dimension" of a lattice polytope. As evidence in favour of this interpretation we present recent results and ongoing work on a conjecture of Batyrev, with an elementary proof in the case of a lattice simplex.

Letzte Aktualisierung: 27.11.2007