We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_{1}, N_{2})$ a basis of $F^{\perp}$ and we set $N = \begin{pmatrix} N_{1} & N_{2} \end{pmatrix} \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$, and $A_{N} = \begin{pmatrix} A & N \\ N^{\top} & 0_{2} \end{pmatrix}$.
Deduce that $\operatorname{det}(A_{N}) = \operatorname{det}(N^{\top}A^{-1}N)\operatorname{det}(A)$.

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. If $1 \leqslant p \leqslant n$, we denote by $\mathcal{G}_{n,p}(\mathbb{R})$ the set of matrices in $\mathcal{M}_{n,p}(\mathbb{R})$ with rank equal to $p$.
Show that there exists $P \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}(P^{\top}A^{-1}P) = 0$ if and only if there exists $P' \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}(P'^{\top}AP') = 0$.

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. We assume $n \geqslant 3$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Let $N' = \begin{pmatrix} N_{1}' & N_{2}' \end{pmatrix}$.
Show that $$\operatorname{det}(N'^{\top}AN') = \left(N_{1}'^{\top}A_{s}N_{1}'\right)\left(N_{2}'^{\top}A_{s}N_{2}'\right) - \left(N_{1}'^{\top}A_{s}N_{2}'\right)^{2} + \left(N_{1}'^{\top}A_{a}N_{2}'\right)^{2}$$

Let $P , Q \in \mathbb { R } [ X ]$ be nonzero polynomials of respective degrees $p$ and $q$ strictly positive. Show that the linear map $L _ { P , Q }$ defined by $$\left\lvert \, \begin{array} { c c c } L _ { P , Q } : \quad \mathbb { R } _ { q - 1 } [ X ] \times \mathbb { R } _ { p - 1 } [ X ] & \rightarrow \quad \mathbb { R } _ { p + q - 1 } [ X ] \\ ( V , W ) & \mapsto V P + W Q \end{array} \right.$$ is an isomorphism if and only if $P$ and $Q$ are coprime in $\mathbb { R } [ X ]$.

Let $d \in \mathbb { N } ^ { * }$. Construct a map $$\left\lvert \, \begin{aligned} r : \quad \mathbb { R } _ { d } [ X ] & \rightarrow \mathbb { R } \\ P & \mapsto r ( P ) \end{aligned} \right.$$ polynomial in the coefficients of $P$, such that, if $r ( P )$ is nonzero, then the roots of $P$ in $\mathbb { C }$ are simple.
Hint: You may use the previous question.

Let $d \in \mathbb { N } ^ { * }$ and $f$ a polynomial function on $\mathbb { R } ^ { d }$. Suppose that the function $f$ is nonzero. Show that $f ^ { - 1 } ( \mathbb { R } \backslash \{ 0 \} )$ is dense in $\mathbb { R } ^ { d }$.
Hint: You may use the fact that a nonzero polynomial in one variable has only finitely many roots.

Let $A \in \mathcal{M}_{n}(\{-1,1\})$. Show that the following statements are equivalent:
(a) $n^{2} \in S(A)$.
(b) There exist $X$ and $Y$ in $\{-1,1\}^{n}$ such that $A = X\,{}^{t}Y$.
(c) $A$ is a rank 1 matrix.

Deduce the proportion, among matrices of $\mathcal{M}_{n}(\{-1,1\})$, of matrices $A$ that satisfy $n^{2} \in S(A)$.

We recall the notation $\underline{M}(n) = \min\left\{M(A) \mid A \in \mathcal{M}_{n}(\{-1,1\})\right\}$. Show that for all $n \geqslant 1$, we have $$\underline{M}(n) \leqslant 2\sqrt{\ln 2}\, n^{3/2}.$$
Hint: one may begin by showing that for all $\varepsilon > 0$, there exists a matrix $A$ in $\mathcal{M}_{n}(\{-1,1\})$ such that $$M(A) \leqslant (2\sqrt{\ln 2} + \varepsilon)\, n^{3/2}.$$

(a) Show that $$\underline{M}(n) \geqslant \frac{n^{2}}{2^{n-1}} \binom{n-1}{\left\lfloor \frac{n}{2} \right\rfloor}.$$
(b) Show next, using Stirling's formula recalled in the preamble, that this lower bound is equivalent to $C n^{\alpha}$ as $n$ tends to infinity, for constants $C$ and $\alpha > 0$ that one will make explicit. Compare with the upper bound for $\underline{M}(n)$ obtained in question 6 of Part II.

We fix $A \in \mathcal{M}_{n}(\{-1,1\})$ and denote $$m(A) := \min(S(A) \cap \mathbb{N}).$$
For $Y \in \{-1,1\}^{n}$, show that we have $$\min\left\{\left|{}^{t}X A Y\right| \mid X \in \{-1,1\}^{n}\right\} \leqslant n$$ and deduce $m(A) \leqslant n$.

Deduce that if $u$, $v$ and $v'$ in $E$ satisfy $v \neq v'$ and $\|u - v\| = \|u - v'\|$ then $\left\|u - \frac{v + v'}{2}\right\| < \|u - v\|$.

Let $F$ be a non-empty closed set of $E$ and $u$ in $E$. Show that there exists $v$ in $F$ such that
$$\forall w \in F, \quad \|u - v\| \leqslant \|u - w\|$$

Deduce that if $C$ is a non-empty closed convex set of $E$ and $u$ is a vector of $E$, then there exists a unique $v$ in $C$ such that
$$\forall w \in F, \quad \|u - v\| \leqslant \|u - w\|$$
We say that $v$ is the projection of $u$ onto $C$ and we denote $d(u, C) = \|u - v\|$.

Let $p$ and $q$ be two strictly positive reals such that $\frac{1}{p} + \frac{1}{q} = 1$. Show that, for all non-negative reals $a$ and $b$,
$$ab \leqslant \frac{a^{p}}{p} + \frac{b^{q}}{q}$$
You may use the concavity of the logarithm.

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Show that, for all non-zero natural integer $p$, there exist two polynomials $P _ { p }$ and $Q _ { p }$ with real coefficients such that, for all $x \in ] - \infty , 1 [$, $$\varphi ^ { ( p ) } ( x ) = \frac { P _ { p } ( \sqrt { 1 - x } ) } { Q _ { p } ( \sqrt { 1 - x } ) } \exp \left( \frac { - x } { \sqrt { 1 - x } } \right)$$

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Deduce $\lim _ { \substack { x \rightarrow 1 \\ x < 1 } } \varphi ^ { ( p ) } ( x )$ for $p \in \mathbb { N } ^ { * }$.

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Deduce that $\varphi$ is of class $C ^ { \infty }$ on $\mathbb { R }$ and for $p \in \mathbb { N } ^ { * }$, give the value of $\varphi ^ { ( p ) } ( 1 )$.

Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and by $\pi$ the orthogonal projection onto $E'$
$$\pi : \left\lvert \, \begin{aligned} E & \rightarrow E' \\ \sum_{i=1}^{n} x_{i} e_{i} & \mapsto \sum_{i=1}^{n-1} x_{i} e_{i} \end{aligned} \right.$$
We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $H_{t}$ the affine hyperplane $E' + te_{n}$ and $C_{t} = \pi(C \cap H_{t})$.
Show that $C_{+1}$ and $C_{-1}$ are non-empty closed convex sets of $E'$.

Show that $\alpha$ has degree 1 if and only if $\alpha \in \mathbb { Q }$, where the degree of $\alpha$ is the degree of its minimal polynomial $\Pi_{\alpha}$.

(a) Show that $\Pi _ { \alpha }$ is irreducible in $\mathbb { Q } [ X ]$.
(b) Let $P \in \mathbb { Q } [ X ]$ be a monic polynomial, irreducible in $\mathbb { Q } [ X ]$. Show that if $z$ is a complex root of $P$, then $P$ is the minimal polynomial of $z$.

(a) Let $A , B \in \mathbb { Q } [ X ]$ be two polynomials that have a common root in $\mathbb { C }$. Show that $A$ and $B$ are not coprime in $\mathbb { Q } [ X ]$.
(b) Show that the roots of $\Pi _ { \alpha }$ in $\mathbb { C }$ are simple.

(a) Show that if $\alpha \in \mathbb { Q }$ is an algebraic integer, then $\alpha \in \mathbb { Z }$.
(b) Show that if $\alpha \in \mathbb { C }$ is an algebraic integer then $\Pi _ { \alpha } \in \mathbb { Z } [ X ]$.
Hint: use the theorem admitted in the introduction (the set of algebraic integers is a subring of $\mathbb{C}$) as well as question 5a.

(a) Let $\alpha \in \mathbb { C }$ be an algebraic integer of degree 2 and of modulus 1. Show that $\alpha$ is a root of unity.
(b) Show that $\frac { 3 + 4 i } { 5 }$ is an algebraic number of degree 2 and of modulus 1 but is not a root of unity.

Show that for all $n \geq 1$ we have $$X ^ { n } - 1 = \prod _ { d \mid n } \Phi _ { d }$$ the product being taken over the set of positive integers $d$ dividing $n$, where $\Phi_n = \prod_{z \in \mathbb{P}_n}(X - z)$ and $\mathbb{P}_n$ is the set of primitive $n$-th roots of unity.