We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ For $x \in \mathbb { R } _ { + } ^ { * }$, we set $\Phi ( x ) = \frac { 4 \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x } - \int _ { 0 } ^ { x } \frac { \mathrm { e } ^ { t / 2 } } { \sqrt { t } } \mathrm {~d} t$. Show that $\Phi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$, that $\lim _ { x \rightarrow 0 } \Phi ( x ) = 0$ and that, for all $x > 0 , \Phi ^ { \prime } ( x ) \geqslant 0$. Deduce that $\Phi ( x ) \geqslant 0$ for all $x > 0$.

We say that the element $M$ of $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$ if, for every $(i,j)$ in $\{1,\ldots,n\}^{2}$, there exists an element $P_{M,i,j}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad \left(R_{z}(M)\right)_{i,j} = \frac{P_{M,i,j}(z)}{\chi_{M}(z)}$$
We admit that every matrix in $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$. Deduce that, if $M \in \mathcal{M}_{n}(\mathbf{C})$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$, there exists an element $P_{M,X,Y}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad X^{T}R_{z}(M)Y = \frac{P_{M,X,Y}(z)}{\chi_{M}(z)}.$$

For every integer $k\geq 1$, recall that $$P_k = \{v\in\mathcal{H}\cap V_\mathbb{Q} \text{ such that } kv\in V_\mathbb{Z}\}.$$ Show that the set $P_k$ is invariant under $\Gamma$.

For all $s>1$, we denote by $P_k(s)$ the subset of $P_k$ formed by vectors $v$ such that $z_v\leq s$.
Show that $P_k(s)$ is finite.

Show that there exists a constant $C>0$ such that for all $v\in\mathcal{H}$, $$|\{g\in\Gamma \text{ such that } gv\in T\}| \leq C.$$

For all $R\in\mathbb{R}$, set $$\Gamma(R) = \{g\in\Gamma \text{ such that } d(v_0,gv_0)\leq R\}.$$ Recall that $\Gamma(R)$ is a finite set. Let $D = \sup_{v\in T} d(v_0,v)$.
Show that, for all $s\geq 0$, $$\frac{1}{C}|\Gamma(\operatorname{arcch}(s)-D)|\cdot|P_k\cap T| \leq |P_k(s)| \leq |\Gamma(\operatorname{arcch}(s)+D)|\cdot|P_k\cap T|.$$

Let $F:[0,2\pi]\times\mathbb{R}_+\rightarrow\mathcal{H}$ be the map defined by $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ For all $(\theta,\alpha)\in[0,2\pi]\times\mathbb{R}_+$ show that $$d\left(F(t,\theta), F\left(t,\theta+\alpha e^{-t}\right)\right) \underset{t\rightarrow+\infty}{\longrightarrow} \operatorname{arcch}\left(1+\frac{\alpha^2}{8}\right)$$ and that the convergence is uniform on every compact subset of $[0,2\pi]\times\mathbb{R}_+$.

For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Show that there exists $r>0$ satisfying the following two properties:

1. for all $g\in\Gamma(n\ln(2))$, there exists $v\in\Delta(n)$ such that $d(gv_0,v)\leq r$,
2. for all $v\in\Delta(n)$, there exists $g\in\Gamma(n\ln(2))$ such that $d(gv_0,v)\leq r$.

For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Fix $r>0$ as in question 6.6. Show that there exists a constant $A\geq 1$ satisfying the following two properties:

1. for all $g\in\Gamma(n\ln(2))$, $$|\{v\in\Delta(n) \text{ such that } d(gv_0,v)\leq r\}| \leq A,$$
2. for all $v\in\Delta(n)$, $$|\{g\in\Gamma(n\ln(2)) \text{ such that } d(gv_0,v)\leq r\}| \leq A.$$

Show the existence of constants $C_1 > C_2 > 0$ and $R_0 > 0$ such that, for all $R\geq R_0$, $$C_2 e^R \leq |\Gamma(R)| \leq C_1 e^R.$$

Deduce the existence of constants $C_1' > C_2' > 0$ and $s_0 > 1$ such that, for all $k\in\mathbb{N}^*$ and all $s\geq s_0$, $$C_2' s\,|P_k\cap T| \leq |P_k(s)| \leq C_1' s\,|P_k\cap T|.$$

Let $H$ be the matrix of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, with general term $h_{i,j} = \phi(X^i, X^j)$. Let $U \in \mathcal{M}_{n,1}(\mathbb{R})$. Express the product $U^\top H U$ using $\phi$ and the coefficients of $U$.

Let $n \in \mathbf { N }$. Show that for all real $t > 0$,
$$p _ { n } = \frac { e ^ { n t } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } P \left( e ^ { - t + i \theta } \right) \mathrm { d } \theta$$
so that
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t + i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta$$

The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u$ is well-defined for all real $t > 0$.

Show that $\int_{1}^{+\infty} \frac{q(u)}{e^{tu}-1} \mathrm{~d}u$ is well defined for all real $t > 0$.

In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
We introduce the function $$\begin{aligned} h : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto h(t) = e^{tA} e^{tB} - e^{t(A+B)} \end{aligned}$$
$\mathbf{7}$ ▷ Show that $$X_k - Y_k = O\left(\frac{1}{k^2}\right) \text{ as } k \rightarrow +\infty.$$

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
Prove that there exists $k \in \llbracket 1; n \rrbracket$ such that $v_{k} \notin F$ and that then $F$ and the vector line spanned by $v_{k}$ are in direct sum.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Let $M \in \mathcal{M}_p(\mathbb{R})$, $e = (e_1, \ldots, e_p)$ and $e^{\prime} = (e_1^{\prime}, \ldots, e_p^{\prime})$ in $E^p$ satisfying $e_i^{\prime} = \sum_{j=1}^p M_{ij} e_j$ for all $i \in \llbracket 1, p \rrbracket$. Show that $\Omega_p(e^{\prime}) = \operatorname{det}(M) \Omega_p(e)$.
(b) Let $e \in E^p$. Show that $\Omega_p(e) \neq 0$ if and only if $e$ is a free family.
(c) Verify that $\Omega_p(e)(e) \geqslant 0$ for all families $e \in E^p$.

If $f, g$ have non-negative real coefficients, $h, g \in O_1$, show that $h \prec g \Rightarrow f \circ h \prec f \circ g$.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
(a) Let $M \in \mathcal{M}_p(\mathbb{R})$, $e = (e_1, \ldots, e_p)$ and $e^{\prime} = (e_1^{\prime}, \ldots, e_p^{\prime})$ in $E^p$ satisfying $e_i^{\prime} = \sum_{j=1}^p M_{ij} e_j$ for all $i \in \llbracket 1, p \rrbracket$. Show that $\Omega_p(e^{\prime}) = \det(M) \Omega_p(e)$.
(b) Let $e \in E^p$. Show that $\Omega_p(e) \neq 0$ if and only if $e$ is a free family.
(c) Verify that $\Omega_p(e)(e) \geqslant 0$ for all family $e \in E^p$. In the sequel for all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = (\det(\operatorname{Gram}(e, e)))^{1/2}$$

Let $C$ be a non-empty closed convex subset of $\mathbb{R}^d$ and let $\sigma_C : \mathbb{R}^d \rightarrow \mathbb{R} \cup \{+\infty\}$ be defined by: $$\sigma_C(p) := \sup\{p \cdot x, x \in C\}$$ show that $$C = \left\{x \in \mathbb{R}^d : p \cdot x \leqslant \sigma_C(p), \forall p \in \mathbb{R}^d\right\}$$ (so that $C$ is an intersection of closed half-spaces).

Let $C$ be a non-empty closed convex subset of $\mathbb{R}^d$ and let $\sigma_C : \mathbb{R}^d \rightarrow \mathbb{R} \cup \{+\infty\}$ be defined by: $$\sigma_C(p) := \sup\{p \cdot x, x \in C\}$$ show that $$C = \left\{x \in \mathbb{R}^d : p \cdot x \leqslant \sigma_C(p), \forall p \in \mathbb{R}^d\right\}$$ (so that $C$ is an intersection of closed half-spaces).

For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. Show that the restriction map $r _ { F } : \begin{array} { c c c } \mathcal { L } ( E , \mathbb { R } ) & \rightarrow & \mathcal { L } ( F , \mathbb { R } ) \\ \ell & \mapsto & \left. \ell \right| _ { F } \end{array}$ is surjective.

For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. Show that the restriction map $r _ { F } : \begin{array} { c c c } \mathcal { L } ( E , \mathbb { R } ) & \rightarrow & \mathcal { L } ( F , \mathbb { R } ) \\ \ell & \mapsto & \left. \ell \right| _ { F } \end{array}$ is surjective.

Show, if $f$ and $g \in O_1$ have non-negative real coefficients and if $r \in [0, \infty]$, that $f \circ g(r) = f(g(r))$.