Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^m\right)$ and $\left(1, (X-1), \ldots, (X-1)^m\right)$ are bases of $\mathbb{R}_m[X]$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_m[X] & \longrightarrow & \mathbb{R}_m[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^m\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^m\right)$ at the end.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$.
Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$\left\{\begin{array}{l} s_1(P) = P(-X) \\ s_2(P) = P(1-X) \\ g(P) = P(X+1) - P(X) \end{array}\right.$$ We thus define three endomorphisms of the vector space $\mathbb{C}_{n-1}[X]$.
Calculate $s_1^2$, $s_2^2$ and express $s_1 \circ s_2$ in terms of $g$ and $Id_{\mathbb{C}_{n-1}[X]}$.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$g(P) = P(X+1) - P(X)$$
Let $P$ be a non-constant polynomial. Express the degree of the polynomial $g(P)$ in terms of the degree of $P$.

Let $E$ be a $\mathbf{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$. Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$g(P) = P(X+1) - P(X)$$ Let $P$ be a non-constant polynomial. Express the degree of the polynomial $g(P)$ in terms of the degree of $P$.

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$, we define the rational function $g_j \in E$ by $$g_j = \frac{f_j}{\prod_{i=1}^{n}(1 - \alpha_i X)}$$ and the map $P_j$, which associates to a rational function $f \in E$ the rational function $$P_j(f) = \frac{(1 - \alpha_j X)f - (1 - \alpha_j^2)f(\alpha_j)}{X - \alpha_j}$$
Show that for every $j \in \llbracket 1, n \rrbracket$, the map $P_j$ is an endomorphism of $E$ and determine its kernel.

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$ and every $g \in E$, compute $P_j\left(\frac{(X - \alpha_j)g}{1 - \alpha_j X}\right)$.

Until the end of part B, we assume that no root of $p$ is stable.
Deduce that the family $(f_1, \ldots, f_n)$ is linearly independent.

Show that the family $\left((S^\top)^i U\right)_{0 \leq i \leq n-1}$ is a basis of $\mathcal{M}_{n,1}(\mathbf{R})$. The matrices $S$ and $U$ were defined in the preliminary part of the problem.

Show, using questions 9 and 13, that if $p$ has no stable root and if $J(p)$ is not invertible then there exists a non-zero polynomial $q$ with real coefficients of degree at most $n-1$ such that $q(S^\top) U = 0_{n,1}$.

A matrix invariant For a square matrix $M$ and a nonzero natural integer $k$, we denote $$\delta_k(M) = -\operatorname{dim}\ker M^{k-1} + 2\operatorname{dim}\ker M^k - \operatorname{dim}\ker M^{k+1}.$$
a) Prove that if two square matrices $M$ and $M'$ are similar, then $\delta_k(M) = \delta_k(M')$ for all $k$.
b) Let $r$ be a nonzero natural integer. Verify that for all nonzero integer $k$, $\delta_k(J_r)$ equals 1 if $k = r$ and 0 otherwise.
c) Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$, $$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$ You may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.

The linear application $\widehat{\xi}$ and the endomorphism $\xi$ We denote by $\widehat{\xi} : \mathbb{C}[X^{\pm 1}] \rightarrow \mathcal{D}$ the linear application that to a Laurent polynomial $F$ associates $$\widehat{\xi}(F) = \Pi(XF) \quad \text{and} \quad \xi = \widehat{\xi}_{\mathcal{D}}$$ that is the endomorphism of $\mathcal{D}$ induced by $\widehat{\xi}$.
a) Let $F$ be an element of $\mathbb{C}[X^{\pm 1}]$. Prove that $\widehat{\xi}(\Pi(F)) = \widehat{\xi}(F)$.
b) Let $P$ be a polynomial and let $F$ be an element of $\mathcal{D}$. Prove that $P(\xi)(F) = \Pi(PF)$.

Image and kernel of powers of $\xi$ Let $n$ be a natural integer. Prove that $\xi^n$ is surjective and give a basis of the kernel of $\xi^n$.

Cyclic subspaces Let $r$ be a nonzero natural integer. Prove that the smallest vector subspace $\mathcal{D}_r$ of $\mathcal{D}$ containing $X^{-r}$ and stable by $\xi$ admits as basis $(X^{k-r})_{0 \leqslant k \leqslant r-1}$. Write the matrix of the endomorphism $\xi_{\mathcal{D}_r}$ induced by $\xi$ on $\mathcal{D}_r$ in this basis.

Compatible extension with $u$ given by a vector Let $V$ be a finite-dimensional vector space equipped with a nilpotent endomorphism $u$. We assume that there exists a vector subspace $W$ of $V$ stable by $u$ and a linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. In this question, we assume that $W$ is strictly contained in $V$ and we fix a vector $v$ of $V$ that does not belong to $W$.
a) Verify that the set $$\mathcal{J} = \{P \in \mathbb{C}[X],\, P(u)(v) \in W\}$$ is an ideal of $\mathbb{C}[X]$.
b) Prove that there exists a natural integer $n$ such that $X^n \in \mathcal{J}$. Deduce that $\mathcal{J}$ is generated by the monomial $X^r$ for an appropriate natural integer $r$ that we do not ask you to specify.
c) Let $W'$ be the subspace of $V$ defined by $$W' = \{P(u)(v) + w,\, P \in \mathbb{C}[X] \text{ and } w \in W\}.$$ Verify that $W'$ contains $W$ and $v$ and that it is stable by $u$.
We denote $G_v = \varphi(u^r(v))$.
d) Prove that there exists an element $F_v$ of $\mathcal{D}$ such that $$G_v = \xi^r(F_v).$$
e) Let $P$ be a polynomial and let $w$ be an element of $W$. Prove that if $P(u)(v) = w$, then $P(\xi)(F_v) = \varphi(w)$.
f) Let $x$ be an element of $W'$. Let $P$ be a polynomial and let $w$ be an element of $W$ such that $x = P(u)(v) + w$. Prove that the element $\varphi'(x) = P(\xi)(F_v) + \varphi(w)$ depends only on $x$ and not on the choice of $P$ and $w$. Verify then that the application $\varphi'$ thus defined is an extension of $\varphi$ to $W'$ compatible with $u$ (it is not asked to verify that $\varphi'$ is linear, which we will admit).

Extension to $V$ compatible with $u$ Let $V$ be a finite-dimensional vector space equipped with a nilpotent endomorphism $u$. We assume that there exists a vector subspace $W$ of $V$ stable by $u$ and a linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. Prove that $\varphi$ admits an extension $\psi$ to $V$ compatible with $u$.

Splitting of a maximal cyclic subspace Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ and let $u$ be an endomorphism of $V$. We assume that $u$ is nilpotent of index $n$, that is $u^n = 0$ and $u^{n-1} \neq 0$. We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero.
a) Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis.
b) Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$.
c) Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.

Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.

Decomposition theorem: uniqueness of block sizes Prove that the number $s$ and the sizes of the blocks $r_1, \ldots, r_s$ that appear in question $9^\circ$ depend only on $u$ and not on the choice of basis. You may use question $2^\circ$.

Conversely, let $K \in \mathcal{M}_n(\mathbb{R})$ be a square matrix of rank 1. Show that there exist $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$ such that $K = \mathbf{u v}^T$.

Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Show that $P$ has a finite number of faces and at least one vertex.

Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Let $V$ be the set of vertices of $P$. Show that $P = \operatorname{Conv}(V)$.

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. Justify that to prove that $\operatorname{Conv}(V)$ is a polytope it suffices to treat the case where $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior.