For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Show that if $x \in \mathbb { C }$ is a root of $P _ { n }$, then $\frac { 1 } { x }$ is also a root of $P _ { n }$, with the same multiplicity.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $t _ { n } + s _ { n } = 6 + n$ and $t _ { n } s _ { n } = 8 + n$.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $s _ { n }$ is real and that $0 < s _ { n } < 2$. Deduce that $\gamma _ { n }$ is not real and that $\gamma _ { n }$ has modulus 1.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
(a) Show that $t _ { n }$ and $s _ { n }$ are irrational.
(b) Deduce that $P _ { n }$ is irreducible in $\mathbb { Q } [ X ]$ and that $\alpha _ { n } \in \mathcal { S }$.
(c) Show that $\lim _ { n \rightarrow + \infty } \alpha _ { n } = + \infty$.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$. Let $\mathcal { T }$ be the set of $\alpha \in \mathcal { S }$ of degree 4. Show that $\mathcal { T }$ has a smallest element and calculate this number.

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 $Q ( X ) = X ^ { d } P \left( \frac { 1 } { X } \right)$. Show that: $$\begin{aligned} Q ( X ) & = 1 + a _ { d - 1 } X + \cdots + a _ { 1 } X ^ { d - 1 } + a _ { 0 } X ^ { d } \\ & = \left( 1 - \lambda _ { 1 } X \right) \left( 1 - \lambda _ { 2 } X \right) \cdots \left( 1 - \lambda _ { d } X \right) \end{aligned}$$

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 $Q ( X ) = X ^ { d } P \left( \frac { 1 } { X } \right)$. Show that: $$\begin{aligned} Q ( X ) & = 1 + a _ { d - 1 } X + \cdots + a _ { 1 } X ^ { d - 1 } + a _ { 0 } X ^ { d } \\ & = \left( 1 - \lambda _ { 1 } X \right) \left( 1 - \lambda _ { 2 } X \right) \cdots \left( 1 - \lambda _ { d } X \right) \end{aligned}$$

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 $\mathbb { R } _ { + }$).
Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

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 $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.

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 $\mathbb { R } _ { + }$).
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 consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ We denote by $\mathbb{C}_n[X]$ the $\mathbb{C}$-vector space of polynomials with complex coefficients and degree at most $n$. Prove that the family $(A_0, \ldots, A_n)$ is a basis of $\mathbb{C}_n[X]$.

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Prove that for all $k \in \llbracket 1, n \rrbracket$, $A_k'(X) = A_{k-1}(X - a)$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda \in \mathbb{C}$, $P_\lambda(X) = P(\lambda X) - P(\lambda)$. For all $\lambda$ in $\mathbb{C}$, $Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$. We consider the polynomial $R(X) = X^{2n} + 1$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Using formula (I.1), show that $$\forall \lambda \in \mathbb{C}, \quad Q_\lambda(X) = -\frac{1}{2n} \sum_{k=1}^{2n} \frac{P(\lambda\omega_k) - P(\lambda)}{\omega_k - 1} \frac{X^{2n} + 1}{X - \omega_k} \omega_k$$ then deduce that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1 - \omega_k)^2} - \frac{P(\lambda)}{2n} \sum_{k=1}^{2n} \frac{2\omega_k}{(1 - \omega_k)^2}.$$

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda$ in $\mathbb{C}$, let $Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X-1} \in \mathbb{C}_{2n-1}[X]$. We consider the polynomial $R(X) = X^{2n} + 1$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Using formula $$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)} \tag{I.1}$$ show that $$\forall \lambda \in \mathbb{C}, \quad Q_\lambda(X) = -\frac{1}{2n} \sum_{k=1}^{2n} \frac{P(\lambda\omega_k) - P(\lambda)}{\omega_k - 1} \frac{X^{2n}+1}{X - \omega_k} \omega_k$$ then deduce that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1-\omega_k)^2} - \frac{P(\lambda)}{2n} \sum_{k=1}^{2n} \frac{2\omega_k}{(1-\omega_k)^2}.$$

Let $n \in \mathbb{N}$ and let $P_n \in \mathbb{R}[X]$ be such that, for all $\theta \in \mathbb{R}$, $\sin((2n+1)\theta) = \sin(\theta) P_n\left(\sin^2(\theta)\right)$.
Determine the roots of $P_n$ and deduce that, for all $x \in \mathbb{R}$, $$P_n(x) = (2n+1) \prod_{k=1}^{n}\left(1 - \frac{x}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right).$$

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Show that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1 - \omega_k)^2} + nP(\lambda)$$ One may apply equality (I.2) to the polynomial $X^{2n}$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Show that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1-\omega_k)^2} + nP(\lambda) \tag{I.2}$$ One may apply equality (I.2) to the polynomial $X^{2n}$.

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ For $k \in \llbracket 1 , n \rrbracket$, we set $y _ { k , n } = \cos \left( \frac { ( 2 k - 1 ) \pi } { 2 n } \right)$. Show that $$T _ { n } ( X ) = 2 ^ { n - 1 } \prod _ { k = 1 } ^ { n } \left( X - y _ { k , n } \right).$$

Let $\alpha \in \mathbf{R}$. Prove that if $\alpha$ is a root of a polynomial $P$ in $\mathbf{R}[X]$ with strictly positive coefficients, then $\alpha < 0$.

Let $n \in \mathbb{N}^*$ and $W$ be a monic polynomial of degree $n$. The objective of this subsection is to show that $$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }.$$ Show that $\sup _ { x \in [ - 1,1 ] } \left| T _ { n } ( x ) \right| = 1$. Deduce a monic polynomial of degree $n$ achieving equality in the above inequality.