For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Deduce the Block and Thielmann theorem: if $(F_n)_{n \in \mathbb{N}}$ satisfies (III.1), then there exists $U \in G$ such that $$\forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ X^n \circ U \quad \text{or} \quad \forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ T_n \circ U$$

We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Prove that $n$ is an element of $\{1, 2, 4, 8\}$.

Let $(A, +, \times)$ be a commutative ring. For $p \in \mathbb{N}^*$, we denote by $C_p(A)$ the set of sums of $p$ squares of elements of $A$. Prove that for every ring $A$, the sets $C_p(A)$ are stable under multiplication when $p$ equals $1, 2, 4$ or $8$.
You may use the bilinear forms $B_p$ defined in part III and, if necessary, restrict yourself to the case where the ring $A$ is the ring $\mathbb{Z}$ of integers.

We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q \in \mathbb{H}$, $N(q) = x^2 + y^2 + z^2 + t^2$ where $q = xe + yI + zJ + tK$. a) Show that $\mathbb{G}$ is a subgroup of $\mathbb{H}$ for addition and that it is stable under multiplication. b) Show that for every $q \in \mathbb{H}$, there exists $\mu \in \mathbb{G}$ such that $N(q - \mu) \leqslant 1$. c) What is the set of $q \in \mathbb{H}$ such that $\forall \mu \in \mathbb{G}, N(q - \mu) \geqslant 1$?

Does the matrix $\Delta _ { p + 1 }$ belong to the set $O ( 1 , p )$ ? to the set $O ^ { + } ( 1 , p )$ ?

Show that $O ( 1 , p ) = O ^ { + } ( 1 , p ) \cup O ^ { - } ( 1 , p )$.

Show that, for every matrix $L$ element of $O ( 1 , p )$, its transpose ${ } ^ { t } L$ is also an element of $O ( 1 , p )$.

Show that the sets $O ( 1 , p ) , O ^ { + } ( 1 , p )$ and $O ^ { - } ( 1 , p )$ of $\mathcal { M } _ { p + 1 } ( \mathbb { R } )$ are closed.

We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Let $L \in \mathcal { M } _ { p + 1 } ( \mathbb { R } )$ and $f$ the endomorphism of $\mathbb { R } ^ { p + 1 }$ canonically associated.
Show that the following three assertions are equivalent:
i. $L \in O ( 1 , p )$;
ii. $\forall \left( v , v ^ { \prime } \right) \in \left( \mathbb { R } ^ { p + 1 } \right) ^ { 2 } , \varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$;
iii. $\forall v \in \mathbb { R } ^ { p + 1 } , q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v )$.

We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ If $L = \left( l _ { i , j } \right) _ { i , j } \in O ( 1 , p ) , v = ( 1,0 , \ldots , 0 )$ and $v ^ { \prime } = ( 0,1,0 , \ldots , 0 )$, give the equations on the $l _ { i , j }$ corresponding to $$\varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right) , \quad q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v ) \quad \text { and } \quad q _ { p + 1 } \left( f \left( v ^ { \prime } \right) \right) = q _ { p + 1 } \left( v ^ { \prime } \right)$$ What do we obtain similarly with ${ } ^ { t } L$ ?

Let $a , b , c$ and $d$ be four real numbers. We consider the matrix of $\mathcal { M } _ { 2 } ( \mathbb { R } )$ $$L = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$$ Write the equations on $a , b , c , d$ expressing the membership of $L$ in $O ( 1,1 )$.

We denote, for every real $\gamma$, $L ( \gamma ) = \left( \begin{array} { c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma \end{array} \right)$.
Show, for all real numbers $\gamma$ and $\gamma ^ { \prime }$, the equality: $$L ( \gamma ) L \left( \gamma ^ { \prime } \right) = L \left( \gamma + \gamma ^ { \prime } \right)$$ Deduce that $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ is a commutative subgroup of the group $O ^ { + } ( 1,1 )$.

Is the group $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ compact?

Show that the group $O ^ { + } ( 1,1 )$ is commutative.

Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ( 1,3 )$. Show the inequality $\ell _ { 1,1 } ^ { 2 } \geqslant 1$.

Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 }$ and $L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$ be two elements of $\tilde { O } ( 1,3 )$. We set $L ^ { \prime \prime } = L L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$.
Prove the following inequalities: $$0 \leqslant \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } ^ { 2 } } \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { k , 1 } ^ { \prime 2 } } + \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } \ell _ { k , 1 } ^ { \prime } < \ell _ { 1,1 } ^ { \prime \prime }$$ Deduce that the set $\tilde { O } ( 1,3 )$ is a subgroup of the Lorentz group $O ( 1,3 )$.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Justify that $G$ is a subgroup of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ isomorphic to $S O ( 3 )$.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$.
Show that, if the vector $a$ is zero, then the matrix $L$ belongs to the group $G$.

Is the decomposition $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ obtained unique?

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Let $M$ and $N$ be two elements of $\mathbf{L}$. Show that $$\exp([M,N]) = \exp(M)\exp(N)\exp(-M)\exp(-N)$$

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and $\mathbf{H} = \{I_3 + M \mid M \in \mathbf{L}\}$, with the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that $\mathbf{H}$ equipped with the usual product of matrices is a subgroup of $\mathrm{SL}_3(\mathbf{R})$ and that $$\exp : (\mathbf{L}, *) \rightarrow (\mathbf{H}, \times)$$ is a group isomorphism.

We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$. We further assume that $A$ and $B$ commute with $[A,B]$.
(a) Show that $[A, \exp(B)] = \exp(B)[A,B]$.
(b) Determine a differential equation satisfied by $t \mapsto \exp(tA)\exp(tB)$.
(c) Deduce the formula: $$\exp(A)\exp(B) = \exp\left(A + B + \frac{1}{2}[A,B]\right)$$

We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$ that commute with $[A,B]$. We denote $\mathcal{L} = \operatorname{Vect}(A, B, [A,B])$.
(a) If $M, N \in \mathcal{L}$, show that $[M,N]$ commutes with $M$ and $N$.
(b) Let $G = \{\exp(M) \mid M \in \mathcal{L}\}$. Show that $(G, \times)$ is a group and that the map $$\Phi : \mathbf{H} \rightarrow G, \quad \exp(M_{p,q,r}) \mapsto \exp(pA + qB + r[A,B])$$ is a group homomorphism.

Let $(D_n)_{n \in \mathbf{N}}$ be a sequence of $\mathcal{M}_d(\mathbf{R})$ that converges to $D \in \mathcal{M}_d(\mathbf{R})$. It is therefore bounded: let $\lambda > 0$ be such that for all integers $n \in \mathbf{N}$, $\|D_n\| \leq \lambda$.
(a) Let $k \in \mathbf{N}$. Justify that $\frac{n!}{(n-k)! n^k} \rightarrow 1$ when $n \rightarrow +\infty$ and that if $n \geq k$ (and $n \geq 1$), $$0 \leq 1 - \frac{n!}{(n-k)! n^k} \leq 1$$ Deduce that $$\left(I_d + \frac{D_n}{n}\right)^n - \sum_{k=0}^{n} \frac{1}{k!}(D_n)^k \rightarrow 0 \quad \text{when } n \rightarrow +\infty$$
(b) Show that for all integers $k \geq 1$ and $n \geq 0$, $$\left\|(D_n)^k - D^k\right\| \leq k\lambda^{k-1}\|D_n - D\|$$
(c) Conclude that $\left(I_d + \frac{D_n}{n}\right)^n \rightarrow \exp(D)$ when $n \rightarrow +\infty$.

Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$.
(a) Let $D \in \mathcal{M}_d(\mathbf{R})$ such that $\|D\| \leq 1$. Show that there exists a constant $\mu > 0$ independent of $D$ such that $$\left\|\exp(D) - I_d - D\right\| \leq \mu \|D\|^2$$
(b) Show that there exists a constant $\nu > 0$, and for all $n \geq 1$ a matrix $C_n \in \mathcal{M}_d(\mathbf{R})$, such that $$\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right) = I_d + \frac{A}{n} + \frac{B}{n} + C_n \quad \text{and} \quad \|C_n\| \leq \frac{\nu}{n^2}$$