For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with $P$ under composition. The Chebyshev polynomials $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Justify that $\mathcal{C}(T_2) = \{-1/2\} \cup \{T_n, n \in \mathbb{N}\}$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P_\alpha)$ the set of complex polynomials that commute with $P_\alpha$ under composition.
Show that the only complex numbers $\alpha$ such that $\mathcal{C}(P_\alpha)$ contains a polynomial of degree three are 0 and $-2$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Deduce the Block and Thielmann theorem: if $(F_n)_{n \in \mathbb{N}}$ satisfies (III.1), then there exists $U \in G$ such that $$\forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ X^n \circ U \quad \text{or} \quad \forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ T_n \circ U$$

We introduce the Dickson polynomials of the first and second kind, $(D_n)_{n \in \mathbb{N}}$ and $(E_n)_{n \in \mathbb{N}}$, defined in the form of polynomial functions of two variables by $$D_0(x,a) = 2 \quad D_1(x,a) = x \quad E_0(x,a) = 1 \quad E_1(x,a) = x$$ then, for every integer $n \in \mathbb{N}$, $$D_{n+2}(x,a) = x D_{n+1}(x,a) - a D_n(x,a) \quad \text{and} \quad E_{n+2}(x,a) = x E_{n+1}(x,a) - a E_n(x,a)$$
Justify the following relation with Chebyshev polynomials $$\forall (x,a) \in \mathbb{C}^2, \quad D_n\left(2xa, a^2\right) = 2a^n T_n(x) \quad \text{and} \quad E_n\left(2xa, a^2\right) = a^n U_n(x)$$ as well as the following two relations, valid for every natural integer $n$ and every $(x,a) \in \mathbb{C}^* \times \mathbb{C}$ $$D_n\left(x + \frac{a}{x}, a\right) = x^n + \frac{a^n}{x^n} \quad \text{and} \quad \left(x - \frac{a}{x}\right) E_n\left(x + \frac{a}{x}, a\right) = \left(x^{n+1} - \frac{a^{n+1}}{x^{n+1}}\right)$$

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We define the polynomials $$Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) \quad \text { and } \quad \forall j \in \{ 1 , \ldots , n \} , \quad P _ { j } = \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } .$$ (a) Show that the family $\left( Q _ { 0 } , P _ { 1 } , P _ { 2 } , \ldots , P _ { n } \right)$ is a basis of $\mathbb { R } _ { n } [ X ]$.
(b) Let $j \in \{ 1 , \ldots , n \}$. Verify that $( - 1 ) ^ { j - 1 } P _ { j } \left( \mu _ { j } \right) > 0$.

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 $$Q _ { 1 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } } \quad \text { and, this time, } \quad P _ { j } = \frac { Q _ { 1 } } { X - \mu _ { j } } .$$ Show that $$Q _ { 1 } \wedge Q _ { 1 } ^ { \prime } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } - 1 }$$

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.

Let $M \in S _ { n + 1 } ( \mathbb { R } )$ be a symmetric matrix. We write $M$ in block form $$M = \left[ \begin{array} { l l } A & y \\ { } ^ { t } y & a \end{array} \right]$$ with $a \in \mathbb { R } , y \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $A \in S _ { n } ( \mathbb { R } )$.
(a) If the spectrum of $A$ is $\operatorname { Sp } ( A ) = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right)$, show that there exist $U \in O _ { n + 1 } ( \mathbb { R } )$ and $z \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ such that $$U M ^ { t } U = \left[ \begin{array} { c c } \Delta \left( \mu _ { 1 } , \ldots , \mu _ { n } \right) & z \\ t _ { z } & a \end{array} \right]$$ (b) Deduce that there exist non-negative real numbers $\alpha _ { j }$ (for $j = 1 , \ldots , n$ ) such that $$\chi _ { M } = ( X - a ) Q _ { 0 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } , \quad \text { where } \quad Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) .$$ (c) Show that $\operatorname { Sp } ( M )$ and $\operatorname { Sp } ( A )$ are interlaced.

We still assume that $p$ is an integer greater than or equal to 2. Let $R \in \mathbb { C } [ X ]$ be a polynomial of degree greater than or equal to 1 with simple roots. Prove that the polynomial $R \left( X ^ { p } \right)$ has simple roots if and only if $R ( 0 ) \neq 0$.

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 $B_{N}$ (see question 2(c)), and we denote it $R_{N}$.
Show that there exist non-negative integers $r, s, t \geqslant 0$, real numbers $c_{1}, \ldots, c_{r}$ different from $\pm 1$, non-zero reals $\rho_{1}, \ldots, \rho_{s}$ and complex numbers $w_{1}, \ldots, w_{t}$ that are neither real nor purely imaginary, such that $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$

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 $[-1,1]$.

We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$).
Show that $\deg R_{N} = 2n$.

We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$, with all roots in $[-1,1]$).
Show that $R_{N}$ is the square of a polynomial: $R_{N}(X) = U_{N}(X)^{2}$ where $U_{N}(1) = 1$ and $U_{N}(-1) = \pm 1$. What can we say about the parity of $U_{N}$?

We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We assume in this question that $U_{N}$ is even; we thus have $U_{N} \in \Pi_{n}$. In $\Pi_{n}$, the equation $P(1) = 1$ defines an affine subspace denoted $H_{n}$.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
(a) Show that $$\left\|U_{N}\right\|_{2} = \min\left\{\|P\|_{2} \mid P \in H_{n}\right\}$$
(b) Deduce that there exists a real number $\mu$ such that for all integers $0 \leqslant j \leqslant \frac{n}{2}$, we have $\left\langle U_{N}, P_{2j} \right\rangle = \mu$. (One may consider polynomials $P \in H_{n}$ of the form $U_{N} + t\left(P_{2j} - P_{2k}\right)$ with $t \in \mathbb{R}$.)
(c) Express $U_{N}$ in the basis of $P_{2j}$. Deduce that $$\frac{1}{\mu} = \sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}$$
(d) Establish in this case the formula $$a_{N} = \left(\sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}\right)^{-1}.$$

We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We now assume that $U_{N}$ is odd.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
Express $a_{N}$ again in terms of the $g_{\ell}$.

Let $(a_0, \ldots, a_{n-1}) \in \mathbb{C}^n$. Let $\lambda$ be a complex number. By discussing in $\mathbb{C}^n$ the system $C(a_0, \ldots, a_{n-1})X = \lambda X$, show that $\lambda$ is an eigenvalue of $C(a_0, \ldots, a_{n-1})$ if and only if $\lambda$ is a root of a polynomial of $\mathbb{C}[X]$ to be specified.

(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.

Show that: $X^p + \alpha_{p-1}X^{p-1} + \cdots + \alpha_0$ divides the polynomial $\chi_f$.

Show that: $X^p + \alpha_{p-1}X^{p-1} + \cdots + \alpha_0$ divides the polynomial $\chi_f$.

Prove that $\chi_f(f)$ is the zero endomorphism.

Prove that $\chi_f(f)$ is the zero endomorphism.

A monic polynomial of degree $d \geq 1$ $$P = \sum _ { i = 0 } ^ { d } a _ { i } X ^ { i } \in \mathbb { C } [ X ]$$ is called reciprocal if $a _ { i } = a _ { d - i }$ for $0 \leq i \leq d$.
(a) Show that a monic polynomial $P \in \mathbb { C } [ X ]$ of degree $d$ is reciprocal if and only if $X ^ { d } P \left( \frac { 1 } { X } \right) = P$.
(b) Let $P \in \mathbb { C } [ X ]$ be a monic reciprocal polynomial. Show that if $x \in \mathbb { C }$ is a root of $P$, then $x \neq 0$ and $\frac { 1 } { x }$ is also a root of $P$, with the same multiplicity.

If $\alpha$ is an algebraic number with minimal polynomial $\Pi _ { \alpha }$, the complex roots of $\Pi _ { \alpha }$ different from $\alpha$ are called the conjugates of $\alpha$. We denote by $C ( \alpha )$ the set of conjugates of $\alpha$.
Let $x$ be an algebraic number of modulus 1 and such that $x \notin \{ - 1,1 \}$. Show that $\frac { 1 } { x }$ is a conjugate of $x$. Deduce that $\Pi _ { x }$ is reciprocal.