On a theorem of Brion

2006/07/12 by Huettemann, Thomas
#05A19 #52B20 #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.math/0607297

Abstract

We give an elementary geometric re-proof of a formula discovered by Michel Brion as well as two variants thereof. A subset of Rⁿ gives rise to a formal Laurent series with monomials corresponding to lattice points in the set. Under suitable hypotheses, these series represent rational functions. We will prove formulae relating the rational function of a lattice polytope P to the sum of rational functions corresponding to the supporting cones subtended at the vertices of P. The exposition should be suitable for everyone with a little background in topology.

Related

• A degenerate version of Brion's formula
• Polar decomposition and Brion's theorem
• A Closer Look at Chapoton's q-Ehrhart Polynomials
• Lattice points in vector-dilated polytopes
• Rational polytopes with Ehrhart coefficients of arbitrary period