Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Let $\gamma_1, \ldots, \gamma_k$ be a family of vectors in $\mathbb{Z}^n \subset \mathbb{R}^n$ and $$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$ Show that if $\gamma_1, \ldots, \gamma_k$ is a free family, $E_{v + C(\gamma_1, \ldots, \gamma_k)}$ is rational for all $v \in \mathbb{R}^n$.

Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$. Let $$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$
Generalize the previous question in the case where $\gamma_1, \ldots, \gamma_k \in \mathbb{Z}^n$ is a family of vectors not necessarily free but for which there exists a linear form $\ell : \mathbb{R}^n \rightarrow \mathbb{R}$ such that $\ell(\gamma_i) > 0$ for $i = 1, \ldots, k$.
Hint. One may triangulate the polytope $P = \{x \in C(\gamma_1, \ldots, \gamma_k) : \ell(x) = 1\}$.

Verify that the image of $\psi$ is contained in the kernel of $\xi^n$.