For each aggregate $V\in\Delta(u)$, we define the polynomial

$$C(\Delta,u,V)=\sum_{k=0}^{\dim(V)}\frac{k!}{\dim(V)!}C_{k}(\Delta,u,V)z^{\dim(V)-k}\in\mathbb{R}[z]$$

recursively. For $k>\dim(V)$ we set $C_{k}(\Delta,u,V)=0$. For convenience, we set $C(\Delta,u,\emptyset)=0$ for all $\Delta$ and $u$. If $u=0$, then $V=\Delta$ and we define $C_{n}(\Delta,0,\Delta)$ as the Lebesgue measure of $\Delta$ and $C_{k}(\Delta,0,\Delta)=0$, for $k<n$. If $u\neq 0$, we set

(7.2)

$$C_{k}(\Delta,u,V)=-\sum_{F}\frac{\langle u_{F},u\rangle}{\|u\|^{2}}C_{k}(F,\pi_{F}(u),V\cap F),$$

where the sum is over the facets $F$ of $\Delta$.