Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$ Let $Q(X)$ be the reciprocal polynomial of $P(X)$ defined by $Q(X) = X^d P\left(\frac{1}{X}\right)$.
We define the function $f : \mathbb { R } \backslash \left( \mathbb { R } \cap \left\{ \frac { 1 } { \lambda _ { 1 } } , \ldots , \frac { 1 } { \lambda _ { d } } \right\} \right) \rightarrow \mathbb { C }$ by $f ( x ) = \frac { Q ^ { \prime } ( x ) } { Q ( x ) }$.
Show that there exists $r > 0$ such that $f$ is expandable as a power series on $] - r , r [$, and that the power series expansion of $f$ at 0 is written as: $$\forall x \in ] - r , r \left[ , \quad f ( x ) = - \sum _ { n = 0 } ^ { \infty } N _ { n + 1 } x ^ { n } \right.$$

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
8a. Show that if $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$, then $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$.
8b. Conversely, show that if $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$, then $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$.
8c. Deduce that if $\mu _ { 1 } , \ldots , \mu _ { d }$ are complex numbers and if $P ( X ) = \prod _ { i = 1 } ^ { d } \left( X - \mu _ { i } \right)$, then $P ( X ) \in \mathbb { Q } [ X ]$ if and only if $$\forall n \geqslant 1 , \quad \sum _ { i = 1 } ^ { d } \mu _ { i } ^ { n } \in \mathbb { Q }$$

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
8a. Show that if $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$, then $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$.
8b. Conversely, show that if $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$, then $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$.
8c. Deduce that if $\mu _ { 1 } , \ldots , \mu _ { d }$ are complex numbers and if $P ( X ) = \prod _ { i = 1 } ^ { d } \left( X - \mu _ { i } \right)$, then $P ( X ) \in \mathbb { Q } [ X ]$ if and only if $$\forall n \geqslant 1 , \quad \sum _ { i = 1 } ^ { d } \mu _ { i } ^ { n } \in \mathbb { Q } .$$

Let $n \geqslant 1$ and $m \geqslant 1$ be two integers and $\alpha _ { 1 } , \ldots , \alpha _ { n } , \beta _ { 1 } , \ldots , \beta _ { m }$ be complex numbers. We define: $$\begin{aligned} & A ( X ) = \left( X - \alpha _ { 1 } \right) \left( X - \alpha _ { 2 } \right) \cdots \left( X - \alpha _ { n } \right) \\ & B ( X ) = \left( X - \beta _ { 1 } \right) \left( X - \beta _ { 2 } \right) \cdots \left( X - \beta _ { m } \right) \end{aligned}$$
Show that if $A ( X )$ and $B ( X )$ have rational coefficients, then the polynomials $$\prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } \beta _ { j } \right) \text { and } \prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } - \beta _ { j } \right)$$ also have rational coefficients.

Let $n \geqslant 1$ and $m \geqslant 1$ be two integers and $\alpha _ { 1 } , \ldots , \alpha _ { n } , \beta _ { 1 } , \ldots , \beta _ { m }$ be complex numbers. We define: $$\begin{aligned} & A ( X ) = \left( X - \alpha _ { 1 } \right) \left( X - \alpha _ { 2 } \right) \cdots \left( X - \alpha _ { n } \right) \\ & B ( X ) = \left( X - \beta _ { 1 } \right) \left( X - \beta _ { 2 } \right) \cdots \left( X - \beta _ { m } \right) . \end{aligned}$$ Show that if $A ( X )$ and $B ( X )$ have rational coefficients, then the polynomials $$\prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } \beta _ { j } \right) \quad \text { and } \quad \prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } - \beta _ { j } \right)$$ also have rational coefficients.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
11a. Show that the set of totally real numbers is a subfield of $\mathbb { R }$. (One may use the result of question 9.)
11b. Show that the set of totally positive numbers is contained in $\mathbb { R } _ { + }$, is closed under addition and multiplication, and that the inverse of a non-zero totally positive number is totally positive.

We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $x$ be a complex number. Show that $x$ is totally real if and only if $x ^ { 2 }$ is totally positive.

We denote by $\mathscr { R }$ the set of totally real numbers and we admit that there exists a function $t : \mathscr { R } \rightarrow \mathbb { Q }$ satisfying the following two properties: (i) for $x , y \in \mathscr { R }$ and $\lambda , \mu \in \mathbb { Q }$, we have $t ( \lambda x + \mu y ) = \lambda t ( x ) + \mu t ( y )$ (ii) for $x$ totally positive, we have $t ( x ) \geqslant 0$ and the equality is strict if $x \neq 0$.
We consider a non-zero totally real number $z$. By definition, there exists a monic polynomial $Z ( X ) \in \mathbb { Q } [ X ]$ that annihilates $z$. We write $Z ( X )$ in the form: $$Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$$ with $d \in \mathbb { N } ^ { \star }$ and $a _ { i } \in \mathbb { Q }$ for all $i \in \{ 0 , \ldots , d - 1 \}$. We further assume that $Z ( X )$ is chosen so that $d$ is minimal among the degrees of monic polynomials $P ( X ) \in \mathbb { Q } [ X ]$ such that $P ( z ) = 0$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j ) , 1 \leqslant i , j \leqslant d$, equals $t \left( z ^ { i + j } \right)$. For $X , Y \in \mathbb { R } ^ { d }$, we set $B ( X , Y ) = X ^ { T } S Y$.
13a. Show that $B ( X , X ) > 0$ for $X \in \mathbb { Q } ^ { d } , X \neq 0$.
13b. Deduce that the matrix $S$ is invertible.

Show that $\lim_{k \rightarrow +\infty} \frac{\operatorname{tr}\left(A^{k+1}\right)}{\operatorname{tr}\left(A^k\right)} = \rho(A)$.

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Deduce from question 3.17 that: $$\|P\|_{\mathbb{D}} = \|P\|_{\partial\mathbb{D}}$$ One may apply question 3.17 to an element $z \in \mathbb{D}$ such that $|P(z)| = \|P\|_{\mathbb{D}}$.

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$: $$|P(z)| \leq \|P\|_{\partial\mathbb{D}} \max\{1, |z|\}^d.$$ One may apply question 3.18 to the polynomials $P(X)$ and $Q(X) = X^d P\left(X^{-1}\right)$.

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Show that there exist $u$ and $v$ in $\partial\mathbb{D}$ such that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq |\lambda| \cdot \prod_{i=1}^{n+m} \max\left\{\left|u - \alpha_i\right|, \left|v - \alpha_i\right|\right\}.$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Deduce from question 3.20 that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq M(S)$$ where $S$ is the polynomial defined by: $$S(X) = (X-1)^{m+n} P\left(\frac{uX - v}{X - 1}\right)$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $w = \frac{v}{u}$. Show that: $$M(S) \leq \|P\|_{\mathbb{D}} \exp\left(\frac{n+m}{2\pi} \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} - w\right|\right\}\right) d\theta\right)$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $C = \exp\left(\frac{I}{2\pi}\right)$ with: $$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$ Using the previous questions, show that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq C^{n+m} \|P\|_{\mathbb{D}}$$

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with $I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta}-1\right|, \left|e^{i\theta}+1\right|\right\}\right)d\theta$. For each natural integer $k \geq 2$, we set: $$\begin{aligned} & Q_k(X) = \prod_{\zeta \in U}(X - \zeta), \\ & R_k(X) = \prod_{\zeta \in V}(X - \zeta), \end{aligned}$$ where $U$ denotes the set of $k$-th roots of unity $\zeta$ such that $|\zeta - 1| \leq |\zeta + 1|$ and $V$ the set of $k$-th roots of unity that are not in $U$. By bounding below the quotient: $$\frac{\left\|Q_k\right\|_{\mathbb{D}} \left\|R_k\right\|_{\mathbb{D}}}{\left\|Q_k R_k\right\|_{\mathbb{D}}},$$ show that: $$C = \inf\left\{\lambda \in \mathbb{R} \left\lvert\, \begin{array}{c} \forall Q \in \mathbb{C}[X]\backslash\{0\},\quad \forall R \in \mathbb{C}[X]\backslash\{0\}, \\ \|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq \lambda^{\operatorname{deg}(QR)} \|QR\|_{\mathbb{D}} \end{array} \right.\right\},$$ where $\operatorname{deg}(QR)$ denotes the degree of $QR$.

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We recall: $$C_{n,m}^I = \sup\left\{\left.\frac{\|Q\|_I \|R\|_I}{\|QR\|_I}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce from question 4.27 that the quantity $C_{n,m}^I$ does not depend on the segment $I$.

Show that, if $A \in \mathcal { M } _ { n } ( \mathbb { R } )$ is nilpotent, then 0 is an eigenvalue of $A$ and that it is the only complex eigenvalue of $A$.

We consider two distinct complex numbers $\alpha$ and $\beta$. We assume that a matrix $A \in \mathcal{M}_3(\mathbf{C})$ has $\alpha$ as a simple eigenvalue and $\beta$ as a double eigenvalue.
$\mathbf{18}$ ▷ Show that $A$ is similar to a matrix of the form $$T = \left( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & a \\ 0 & 0 & \beta \end{array} \right)$$ where $a$ is a certain complex number. Compute $T^n$ for $n$ a natural integer, then $e^{tT}$ for $t$ real. Deduce from this a necessary and sufficient condition on $\alpha$ and $\beta$ for $\lim_{t \rightarrow +\infty} e^{tA} = 0_3$.

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. For every eigenvalue $\lambda$ of the matrix $A$, we denote by $m_\lambda$ its multiplicity, and we introduce the vector subspace $$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right) = \operatorname{Ker}\left((u - \lambda \operatorname{Id}_E)^{m_\lambda}\right).$$
$\mathbf{20}$ ▷ Show that $\mathbf{C}^n = \bigoplus_{\lambda \in \operatorname{Sp}(A)} F_\lambda$.

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. For every eigenvalue $\lambda$ of the matrix $A$, we denote by $m_\lambda$ its multiplicity, and we introduce the vector subspace $$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right).$$
$\mathbf{21}$ ▷ Deduce from question 20) the existence of three matrices $P, D$ and $N$ in $\mathcal{M}_n(\mathbf{C})$ such that $A = P(D + N)P^{-1}$, where $D$ is diagonal, $N$ is nilpotent, $DN = ND$, and $P$ is invertible.

Consider $(c_0, \ldots, c_{d-1}) \in \left(\mathbb{R}_{+}^{*}\right)^d$ and $P$ the polynomial $$X^d - c_{d-1} X^{d-1} - \cdots - c_1 X - c_0.$$ Show that the polynomial $P$ has a unique root in $\mathbb{R}_{+}^{*}$.

Prove Theorem 1: Let $n \in \mathbb { N }$ be a natural integer and let $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ be monic. Let $\lambda \in \mathbb { R }$ be a root of $P _ { \mid t = 0 }$ of multiplicity $d$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n - d } [ X ] \right)$ monic such that $P = F G$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.

Let $R$ be a non-constant polynomial of $\mathbf{C}[X]$ having the following property: Every root $a$ of $R$ is nonzero and $\frac{1}{a}$ is a root of $R$ with the same multiplicity as $a$.
Prove that the product of the roots of $R$, counted with multiplicities, can only take the values 1 or $-1$. One may note that the equality $a = \frac{1}{a}$ holds only for $a = 1$ or $-1$.

Let $R$ be a non-constant polynomial of $\mathbf{C}[X]$ having the following property: Every root $a$ of $R$ is nonzero and $\frac{1}{a}$ is a root of $R$ with the same multiplicity as $a$. Prove that the product of the roots of $R$, counted with multiplicities, can only take the values 1 or $-1$. One may note that the equality $a = \frac{1}{a}$ holds only for $a = 1$ or $-1$.