## 代数幾何セミナー：On enumeration of lattice points in virtual polytopes

2015年 　 7月 31日 16時 30分 ～ 　 17時 30分

Thomas Huttemann (Belfast)

アブストラクト
Let P be a polytope (in R^n) with integral vertex coordinates. Its integral points define a Laurent polynomial in n variables: the coordinates of the integral points appear as the exponents of the variables. This is the lattice point enumerator of P.

Similarly, any subset S of R^n determines an infinite sum of Laurent monomials, by summing over integral points in the given set. If S is the cone on P subtended by a vertex of P, this series converges to a rational function. A surprising theorem of Michel Brion asserts that the sum of these rational functions corresponding to vertices of P is precisely the lattice point enumerator of P.

In the talk I will explain the theorem in some detail and give an elementary proof. I will then generalise the theorem to so-called virtual polytopes, arriving at a cohomological re-interpretation and generalisation of Brion's original theorem.