Number of integral points in Convex rational Polytopes

Michèle Vergne

Ecole Polytechnique, France

e-mail vergne@math.polytechnique.fr

We all tried in our youth to work out formulae for sums of the m-th powers of the first N numbers, and to compare them to the corresponding integral $\int_0^N x^m dx$. This leads to Bernoulli numbers $B_n$ and to the first appearance of the power series

$$\frac{z}{e^{z}-1}=\sum_{n} B_n \frac{z^n}{n!}.$$

We will then ask ourselves if there are integral formulae computing the number of points with integral coordinates in a convex polytope in $R^n$ given by rational inequalities.

We will see that the solution by Bernoulli, Euler-Mac Laurin,..., Khovanski-Pukhlikhov, ... ,Brion-Vergne, of this problem is similar to the amazing Riemann-Roch theorem: an integral of characteristic classes resulting in a positive integral number.

We will also indicate a number of very simple open problems on number of magic squares, and a few new results due to Chan-Robbins and Zeilberger, and Baldoni-Vergne, of the number of points on some "concrete polytopes" related to magic squares.