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
Abstract:
Colloquium - 16:00
Benjamin Nill - FU Berlin
The degree of lattice polytopes
Abstract:
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.