Ehrhart Theory and Unimodular Decompositions of Lattice Polytopes

Abstract

Ehrhart theory studies the behaviour of lattice points contained in dilates of lattice polytopes. We provide an introduction to Ehrhart theory. In particular, we prove Ehrhart's Theorem, Stanley Non-negativity, and Ehrhart-Macdonald Reciprocity via lattice triangulations. We also introduce the algebra $\mathscr{P}(\mathbb{R}^d)$ spanned by indicator functions of polyhedra, and valuations (linear functions) on $\mathscr{P}(\mathbb{R}^d)$. Through this, we derive Brion's Theorem, which gives an alternate proof of Ehrhart's Theorem. The proof of Brion's Theorem makes use of decomposing the lattice polytope in $\mathscr{P}(\mathbb{R}^d)$ into support cones and other polyhedra. More generally, Betke and Kneser proved that every lattice polytope in $\mathscr{P}(\mathbb{R}^d)$ (or the sub-algebra $\mathscr{P}(\mathbb{Z}^d)$, spanned by lattice polytopes) admits a unimodular decomposition; it can be expressed as a formal sum of unimodular simplices. We give a new streamlined proof of this result, as well as some applications and consequences.