Suppose that for a given polynomial $p ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c x + d$, there is exactly one real number $r$ such that $p ( r ) = 0$.
(a) If $a, b, c, d$ are rational, show that $r$ must be rational.
(b) If $a, b, c, d$ are integers, show that $r$ must be an integer.
Possible hint: Also consider the roots of the derivative $p ^ { \prime } ( x )$.

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Let $n \in \mathbb{N}^*$. Show that the polynomial function $T_n$ has exactly $n$ distinct zeros all belonging to $]-1,1[$. For $j \in \{1, 2, \ldots, n\}$, we denote by $x_{n,j}$ the $j$-th zero of $T_n$ in increasing order. Give the value of $x_{n,j}$.

We assume $\alpha = 1$. We denote $T_n$ the unique polynomial eigenvector of $\varphi_1$ of degree $n$, of norm 1 (with respect to $S_1$) and with positive leading coefficient. For $n \in \mathbb{N}^*$, determine the roots of $T_n$.

The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Show that, for every natural integer $n$, the polynomial $T_n$ is split over $\mathbb{R}$, with simple roots belonging to $]-1,1[$. Determine the roots of $T_n$.

The Chebyshev polynomials of the second kind $(U_n)_{n \in \mathbb{N}}$ are defined by $U_n = \frac{1}{n+1} T_{n+1}'$, and satisfy $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$.
Deduce the following properties:
a) The sequence $(U_n)_{n \in \mathbb{N}}$ satisfies the same recurrence relation $T_{n+2} = 2X T_{n+1} - T_n$ as the sequence $(T_n)_{n \in \mathbb{N}}$.
b) For every natural integer $n$, the polynomial $U_n$ is split over $\mathbb{R}$ with simple roots belonging to $]-1,1[$. Determine the roots of $U_n$.

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We define the polynomials $$Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) \quad \text { and } \quad \forall j \in \{ 1 , \ldots , n \} , \quad P _ { j } = \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } .$$ Let $P \in \mathbb { R } [ X ]$ be a monic polynomial of degree $n + 1$.
(a) Show that there exists a unique vector $\left( a , \alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { n } \right) \in \mathbb { R } ^ { n + 1 }$ such that $$P = ( X - a ) Q _ { 0 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } P _ { j }$$ (b) Assume that the real numbers $\alpha _ { 1 } , \ldots , \alpha _ { n }$ are all strictly positive. Show that $P$ has $n + 1$ distinct real roots $\lambda _ { 1 } > \cdots > \lambda _ { n + 1 }$, and that $\hat { \lambda } = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$ and $\widehat { \mu }$ are strictly interlaced.
(c) Conversely, assume that $P$ has $n + 1$ distinct real roots $\lambda _ { 1 } > \cdots > \lambda _ { n + 1 }$, and that $\widehat { \lambda } = \left( \lambda _ { 1 } > \cdots > \lambda _ { n + 1 } \right)$ and $\widehat { \mu }$ are strictly interlaced. Show that, for all $j \in \{ 1 , \ldots , n \} , \alpha _ { j } > 0$.

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We are given integers $m _ { k } \geqslant 1$ for $k = 1 , \ldots , n$. We set $$Q _ { 1 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } } \quad \text { and } \quad P _ { j } = \frac { Q _ { 1 } } { X - \mu _ { j } } .$$ Let $\left( a , \alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { n } \right) \in \mathbb { R } ^ { n + 1 }$ and let $P \in \mathbb { R } [ X ]$ be defined by the formula $$P = ( X - a ) Q _ { 1 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } P _ { j }$$ (a) Give an expression of $P \wedge Q _ { 1 }$ in terms of the $\mu _ { j }$, the $m _ { j }$ and the set $J$ of indices for which $\alpha _ { j } = 0$.
(b) Assume that the numbers $\alpha _ { 1 } , \ldots , \alpha _ { n }$ are non-negative. Show that all roots of $P$ are real.

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The polynomial $K(X)$ is defined as in question 23.
Deduce that the roots of $K$ are all real and belong to the interval $]0, 4[$.

We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
Using the results of questions 8, 9, and 10, conclude that $R_{N}$ has all its roots in the interval $[-1,1]$.

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 $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ 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 $\mathcal { S }$ 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.

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.

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 $n \in \mathbf{N}^{*}$. Let $(z_{1}, z_{2}, \ldots, z_{n}) \in \mathbf{C}^{n}$. We define the two polynomials $P(X)$ and $Q(X)$ in $\mathbf{C}[X]$ by: $$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$
Prove that if $P$ and $Q$ are in $\mathbf{R}[X]$, then we have the equivalence: $P$ is a Hurwitz polynomial if and only if the coefficients of $P$ and $Q$ are strictly positive.

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Show that there exists a real number $\eta > 0$ such that for every $r \in ]1-\eta; 1[$, the polynomial $p(rX)$ is split, has exactly $\sigma(p)$ roots inside the interval $]-1; 1[$ and has no stable root.