Recall by Theorem 4.6.1 if \(\Delta \subset \mathbb{R}^d\) is a lattice simplex, then for a positive integers \(n\), the counting function \(\operatorname{ehr}_{\Delta}(n)\) is a polynomial in \(n\) of degree \(\dim(\Delta)\) whose constant term equals \(1\). Hence for a general lattice polytope \(P \in \mathbb{R}^d\), we would want to find simplicies \(\Delta_1, \Delta_2, \cdots, \Delta_m\) such that

$$ \begin{align*} P = \Delta_1 \cup \Delta_2 \cup \cdots \cup \Delta_m \end{align*} $$

Then by the exclusion-inclusion formula

$$ \begin{align*} \operatorname{ehr}_P(n) = \sum_{\varnothing \neq I \subseteq [m]} (-1)^{|I|-1} \operatorname{ehr}_{\Delta_{\cap I}}(n) \end{align*} $$

However, \(\Delta_{\cap I}\) is not necessarily a lattice simplex. Hence, we can’t apply 4.6.1 to conclude that \(\operatorname{ehr}_{\Delta \cap I}(n)\) is a polynomial in \(n\). Therefore, we need a way to subdivide a given polytope nicely such that we can still apply Theorem 4.6.1 on the intersections. To do this, we start with a few definitions from chapter 5:

Definition (dissection)
dissection of a polyhedron \(Q \subset \mathbb{R}^d\) is a collection of polyhedra \(Q_1,\ldots,Q_m\) of the same dimension such that $$ \begin{align*} Q &= Q_1 \cup \cdots \cup Q_m, \\ Q_i^\circ \cap Q_j^\circ &= \varnothing \qquad \text{whenever } i \neq j. \end{align*} $$
Definition (polyhedral complex)
polyhedral complex is a nonempty finite collection \(\mathcal S\) of polyhedra in \(\mathbb{R}^d\) (called the cells of \(\mathcal S\)) satisfying
  • Containment property: If \(F\) is a face of \(G \in \mathcal S\), then \(F \in \mathcal S.
  • Intersection property: If \(F,G \in \mathcal S\), then \(F \cap G\) is a face of both \(F\) and \(G\).

A polyhedral complex \(\mathcal S\) is called

  • a polytopal complex if all of its cells are polytopes.
  • a fan if all of its cells are polyhedral cones.
  • a geometric simplicial complex} if all of its cells are simplices.

The support of a polyhedral complex \(\mathcal S\) is

$$ \begin{align*} |\mathcal S| := \bigcup_{F\in\mathcal S} F, \end{align*} $$

the point set underlying \(\mathcal S\). The vertices of \(\mathcal S\) are the zero-dimensional polytopes contained in \(\mathcal S\).

Definition (polyhedral complex)
A subdivision of a polyhedron \(P \subset \mathbb{R}^d\) is a polyhedral complex \(\mathcal S\) such that \(P = |\mathcal S|\).