We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $r$ is $$B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$$ and the symplectic cylinder of radius $r$ is $$Z ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant r ^ { 2 } \right\}.$$ Show that, for all $r > 0$, there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$.

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $r$ is $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$.
Let $r > 0$ such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$. Let $U \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ denote the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { 2 m }$. Let $\lambda \in \mathbb { C }$ be a complex eigenvalue of the matrix $U$. Show that $| \lambda | \leqslant r$.
For the case $\lambda$ non-real, if $P$ and $Q$ in $\mathcal { M } _ { 2 m , 1 } ( \mathbb { R } )$ are such that $Z = P + \mathrm { i } Q$ is an eigenvector column of $U$ for the eigenvalue $\lambda$, one may show that $\| U P \| ^ { 2 } + \| U Q \| ^ { 2 } = | \lambda | ^ { 2 } \left( \| P \| ^ { 2 } + \| Q \| ^ { 2 } \right)$.

Let $r > 0$ be such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$. Let $U \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ denote the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { 2 m }$. Let $\lambda \in \mathbb { C }$ be a complex eigenvalue of the matrix $U$. Show that $| \lambda | \leqslant r$.
For the case $\lambda$ non-real, if $P$ and $Q$ in $\mathcal { M } _ { 2 m , 1 } ( \mathbb { R } )$ are such that $Z = P + \mathrm { i } Q$ is an eigenvector column of $U$ for the eigenvalue $\lambda$, one may show that $\| U P \| ^ { 2 } + \| U Q \| ^ { 2 } = | \lambda | ^ { 2 } \left( \| P \| ^ { 2 } + \| Q \| ^ { 2 } \right)$.

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$, and every complex eigenvalue $\lambda$ of the matrix $U$ of $u$ satisfies $|\lambda| \leq r$. Deduce that $1 \leqslant r$.

Let $r > 0$ be such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$, and let $U$ be the matrix of $u$. Given that every complex eigenvalue $\lambda$ of $U$ satisfies $|\lambda| \leq r$, deduce that $1 \leqslant r$.

We fix $n = 2m \geqslant 4$ and $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$. Under what necessary and sufficient condition on $r > 0$ does there exist $u$ belonging to $\mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$?

Under what necessary and sufficient condition on $r > 0$ does there exist $u$ belonging to $\mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$?

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $r$ is $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$ and the symplectic cylinder is $Z ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant r ^ { 2 } \right\}$.
Let $r > 0$ such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$. We denote by $M \in \operatorname { Sp } _ { 2 m } ( \mathbb { R } )$ the matrix of $\psi$ in the canonical basis $\left( e _ { 1 } , \ldots , e _ { m } , f _ { 1 } , \ldots , f _ { m } \right)$ of $\mathbb { R } ^ { 2 m }$ and by $\psi ^ { \top }$ the endomorphism canonically associated with $M ^ { \top }$. Show that $\left| b _ { s } \left( \psi ^ { \top } \left( e _ { 1 } \right) , \psi ^ { \top } \left( f _ { 1 } \right) \right) \right| = 1$ then that $\left\| \psi ^ { \top } \left( e _ { 1 } \right) \right\| \geqslant 1$ or $\left\| \psi ^ { \top } \left( f _ { 1 } \right) \right\| \geqslant 1$.

Let $r > 0$ be such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$. We denote by $M \in \operatorname { Sp } _ { 2 m } ( \mathbb { R } )$ the matrix of $\psi$ in the canonical basis $\left( e _ { 1 } , \ldots , e _ { m } , f _ { 1 } , \ldots , f _ { m } \right)$ of $\mathbb { R } ^ { 2 m }$ and by $\psi ^ { \top }$ the endomorphism canonically associated with $M ^ { \top }$. Show that $\left| b _ { s } \left( \psi ^ { \top } \left( e _ { 1 } \right) , \psi ^ { \top } \left( f _ { 1 } \right) \right) \right| = 1$ then that $\left\| \psi ^ { \top } \left( e _ { 1 } \right) \right\| \geqslant 1$ or $\left\| \psi ^ { \top } \left( f _ { 1 } \right) \right\| \geqslant 1$.

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$, where $B^{2m}(r)$ is the closed Euclidean ball and $Z^{2m}(r)$ is the symplectic cylinder. Using the result of Q47, show that $1 \leqslant r$.

Let $r > 0$ be such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$. Using the result of Q47, show that $1 \leqslant r$.

Let $A$ be a commutative ring. Let $S _ { 1 }$ and $S _ { 2 }$ be two subsets of $A$ such that $S _ { 1 } \subset \mathcal { A } \left( S _ { 2 } \right)$. Show that $\mathcal { A } \left( S _ { 1 } \right) \subset \mathcal { A } \left( S _ { 2 } \right)$.

Show that every finite abelian group and the additive group $\mathbf { Z } ^ { r }$ for $r \in \mathbf { N } ^ { * }$ have property (F).

Show that if $n$ is a strictly positive integer, the ring $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ has property (TF), but not property (F).

Show that the ring $\mathbf { Q }$ of rational numbers does not have property (TF).

Let $f : A \rightarrow B$ be a morphism of commutative rings. Let $F$ be an element of $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$. Show that we have $f \left( F \left( a _ { 1 } , \ldots , a _ { n } \right) \right) = F \left( f \left( a _ { 1 } \right) , \ldots , f \left( a _ { n } \right) \right)$ for all $a _ { 1 } , \ldots , a _ { n } \in A$.

Let $B$ be a commutative ring. Let $n$ be a strictly positive integer and $b _ { 1 } , \ldots , b _ { n }$ be elements of $B$. a) Show that there exists a unique ring morphism $f$ from $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ to $B$ such that $f \left( X _ { i } \right) = b _ { i }$ for all $i \in \{ 1 , \ldots , n \}$. b) Deduce that $B$ has property (TF) if and only if there exist an integer $n \geq 1$ and a surjective ring morphism $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right] \rightarrow B$. c) Show that an abelian group $M$ has property (F) if and only if there exist an integer $r \geq 1$ and a surjective group morphism $\mathbf { Z } ^ { r } \rightarrow M$. d) Let $A$ and $B$ be commutative rings such that there exists a surjective ring morphism from $A$ to $B$. Show that if $A$ has property (TF), then so does $B$. State and prove an analogous statement for property (F).

Let $M$ be an additive subgroup of $\mathbf { Z } ^ { n }$ with $n \in \mathbf { N }$ (we agree that $\mathbf { Z } ^ { 0 }$ is the trivial group). We propose to prove by induction on $n$ the following result: (*) There exists $r \in \mathbf { N }$ such that the abelian group $M$ is isomorphic to $\mathbf { Z } ^ { r }$. a) Verify the cases $n = 0$ and $n = 1$. We now assume the result is true for $n - 1$. Let $p : \mathbf { Z } ^ { n } \rightarrow \mathbf { Z }$ be the projection onto the first coordinate, we denote by $N$ the kernel of $p$ and $N _ { 1 } = M \cap N$, then we set $p ( M ) = a \mathbf { Z }$ with $a \in \mathbf { Z }$. We choose $e _ { 1 } \in M$ such that $p \left( e _ { 1 } \right) = a$. Show that if $a \neq 0$, then the application $$N _ { 1 } \times \mathbf { Z } \rightarrow M , ( x , m ) \mapsto x + m e _ { 1 }$$ is a group isomorphism. b) Deduce (*). c) Show that the integer $r$ such that $M$ is isomorphic to $\mathbf { Z } ^ { r }$ is unique (one may consider the rank of a family of vectors of $\mathbf { Z } ^ { r }$ in the $\mathbf { Q }$-vector space $\mathbf { Q } ^ { r }$).

Show that if an abelian group $M$ has property (F), then every subgroup of $M$ also has it.

We consider the ring $A = \mathbf { Z } [ X , Y ]$. Let $U$ be the set of elements of $A$ of the form $X Y ^ { k }$ with $k \in \mathbf { N }$, we set $B = \mathcal { A } ( U )$. Let $S$ be a finite subset of $B$. a) Show that there exists $m \in \mathbf { N } ^ { * }$ such that $\mathcal { A } ( S ) \subset \mathcal { A } \left( \left\{ X , X Y , \ldots , X Y ^ { m } \right\} \right)$. b) Show that there exists an integer $N > 0$ such that every element of $\mathcal { A } ( S )$ is a sum of monomials of the form $\alpha X ^ { i } Y ^ { j }$ with $\alpha \in \mathbf { Z }$ and $j \leq i N$. c) Deduce that the ring $B$ does not have property (TF).

Let $E$ be a finite subset of $M _ { n } ( A )$. Show that there exists a subring $B$ of $A$ such that: $B$ has property (TF) and for every matrix $M \in E$, all coefficients of $M$ belong to $B$.

Let $M$ be a matrix of $M _ { n } ( A )$. The purpose of this question is to generalize to an arbitrary commutative ring $A$ the two formulas recalled in the introduction when $A$ is a field. a) Show that if the ring $A$ is integral, then $M \widetilde { M } = \widetilde { M } M = ( \operatorname { det } M ) I _ { n }$. b) We no longer assume $A$ is integral. Show that the result of a) still holds if there exists a surjective ring morphism $B \rightarrow A$ with $B$ integral. c) Deduce that the result of a) still holds for every commutative ring $A$. d) Prove that if $M$ and $N$ are in $M _ { n } ( A )$, then we have $$\operatorname { det } ( M N ) = \operatorname { det } M \times \operatorname { det } N .$$

Let $r$ and $s$ be strictly positive integers. Let $M \in M _ { s , r } ( A )$. We consider the application $u : A ^ { r } \rightarrow A ^ { s }$ defined by $u ( X ) = M X$, where we identify elements of $A ^ { r }$ and $A ^ { s }$ with column vectors. We assume that $u$ is surjective and that the ring $A$ is not reduced to $\{ 0 \}$. The purpose of this question is to prove that we then have $r \geq s$. For this, we reason by contradiction by assuming $r < s$. a) Show that there exists a matrix $N \in M _ { r , s } ( A )$ such that $M N = I _ { s }$. b) We define matrices of $M _ { s } ( A )$ by blocks: $$\begin{aligned} M _ { 1 } & = \left( \begin{array} { l l } M & 0 \end{array} \right) \\ N _ { 1 } & = \binom { N } { 0 } \end{aligned}$$ In other words, $M _ { 1 }$ is the matrix obtained by adding $s - r$ zero columns to $M$ and $N _ { 1 }$ is the matrix obtained by adding $s - r$ zero rows to $N$. Calculate $M _ { 1 } N _ { 1 }$. c) Reach a contradiction and conclude. d) We assume that $r = s$. Show the equivalence of the following properties: i) The application $u$ is surjective; ii) The determinant $\operatorname { det } M$ belongs to $A ^ { * }$; iii) There exists $N \in M _ { r } ( A )$ such that $M N = N M = I _ { r }$. iv) The application $u$ is bijective.

Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We denote by $G L _ { r } ( A )$ the set of matrices of $M _ { r } ( A )$ that satisfy the equivalent properties of question III.3.d). a) Show that matrix multiplication induces a group structure on $G L _ { r } ( A )$. b) We define a relation on $M _ { s , r } ( A )$ by $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$. Show that this is an equivalence relation.

Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We say that two matrices $M$ and $N$ of $M _ { s , r } ( A )$ are $A$-equivalent if $M \sim N$ (where $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$). If $M \in M _ { s , r } ( \mathbf { Z } )$ and $k$ is an integer at most equal to $\min ( r , s )$, we denote by $m _ { k } ( M )$ the gcd of the minors of size $k$ of $M$. Let $M$ and $N$ be two $\mathbf { Z }$-equivalent matrices of $M _ { s , r } ( \mathbf { Z } )$. Show that for all $k \leq \min ( r , s )$, we have $m _ { k } ( M ) = m _ { k } ( N )$ (one may begin by showing that $m _ { k } ( M )$ divides $m _ { k } ( N )$).