Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $S_n$ the symmetric group of order $n$. For all $\sigma \in S_n$, we define $P^\sigma \in \mathcal{M}_n(\mathbb{R})$ as follows: for $i, j \in \{1, 2, \ldots, n\}$ we set $P^\sigma_{ij} = 1$ if $j = \sigma(i)$, $P^\sigma_{ij} = 0$ otherwise. Show that $P^\sigma$ is a vertex of $B_n$ for all $\sigma \in S_n$.

Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$. We say that $f \in \mathbb{C}[[\mathbb{Z}^n]]$ is rational if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf \in \mathbb{C}[\mathbb{Z}^n]$. We say that $f$ is torsion if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf = 0$. We denote by $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements and $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements of $\mathbb{C}[[\mathbb{Z}^n]]$.
In the case where $n = 1$, show that the inclusions $0 \subset \mathcal{T} \subset \mathcal{R} \subset \mathbb{C}[[\mathbb{Z}^n]]$ are strict.

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 $P = \{x \in C(\gamma_1, \ldots, \gamma_k) : \ell(x) = 1\}$.

21. Show that the polynomial $H \left( f , \prod _ { i = 1 } ^ { n } \left( X - r _ { i } \right) \right)$ is a function of positive type in dimension $N$.

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

A square made of cardboard has sides of unit length and corners marked $A , B , C , D$.\ (a) Show that there are eight different ways of placing the cardboard square so that it completely covers the square region in the $( x , y )$ plane with corners at the points $( 0,0 ) , ( 0,1 ) , ( 1,1 ) , ( 1,0 )$.\ (b) Initially the square is positioned so that $A , B , C , D$ are over the points $( 0,0 ) , ( 0,1 )$, $( 1,1 ) , ( 1,0 )$, respectively. You may move the square by either\ (i) rotating it in the plane by $90 ^ { \circ }$ about one of the corners, or\ (ii) turning it over keeping one of the edges fixed in contact with the plane.
Show that after two moves it is possible to return the square to its initial position but with the corners $B$ and $D$ interchanged.\ (c) Show that four of the eight configurations in (a) can be achieved from the initial position and moves described in (b).