5:00PM: Counting lattice points and the number 2i+7
Steven Sam
Let P be a lattice d-dimensional polytope. A theorem of E.
Ehrhart states that the number of lattice points of nP (the nth dilate
of P), as a function of positive integers n, agrees with a polynomial of
degree d (the Ehrhart polynomial of P). In this talk, I will give a
review of Ehrhart polynomials and discuss a paper of C. Haase and J.
Schicho that classifies all such polynomials for d=2.
If we replace ``lattice polytope'' by ``rational polytope,'' the word
``polynomial'' becomes ``polynomial with periodic
coefficients'' (quasi-polynomials). I will discuss known bounds for the
periods of Ehrhart quasi-polynomials, but show that in some cases, they
are ordinary polynomials. I will also show that the classification for
d=2 is incomplete if we consider rational polygons whose Ehrhart
quasi-polynomial is a polynomial, and present some conjectures that I
like about why some polytopes have smaller period than the known bounds
suggest.
Attachments
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