Let $P$ be an irreducible Hurwitz polynomial in $\mathbf{R}[X]$ with positive leading coefficient. Prove that all coefficients of $P$ are strictly positive.

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: $$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$
We assume $n = 2$ and $P \in \mathbf{R}_{2}[X]$. If the coefficients of $Q$ are strictly positive, is $P$ then a Hurwitz polynomial?

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.

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 $\mathbb{R}_{+}^{*}$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let $$m = \min\{k \in \mathbb{N} \mid \exists P \in \mathscr{V}(A) \text{ with } \deg(P) = k\}$$ Show that there exists a unique polynomial $p \in \mathbb{C}[X]$ satisfying the three conditions
(i) $p \in \mathscr{V}(A)$,
(ii) $\deg(p) = m$,
(iii) $p$ is monic.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Show that $$\varphi_A(X) = (X - \lambda_1) \cdots (X - \lambda_\ell).$$

Let $P \in \mathbf{C}[X]$ of degree $p$. We write $P = \sum_{k=0}^{p} a_k X^k$, where $a_0, \ldots, a_p$ are complex numbers, and $a_p \neq 0$.
Show that $P$ is reciprocal if and only if for every integer $k$, $0 \leq k \leq p$, we have the equality $a_k = a_{p-k}$.

Let $P \in \mathbf{C}[X]$ of degree $p$. We write $P = \sum_{k=0}^{p} a_k X^k$, where $a_0, \ldots, a_p$ are complex numbers, and $a_p \neq 0$. Show that $P$ is reciprocal if and only if for every integer $k$, $0 \leq k \leq p$, we have the equality $a_k = a_{p-k}$.

Show that $p_0$, the reciprocal polynomial of $p$, satisfies $$\forall x \in \mathbf{R}^* \quad p_0(x) = x^n p(1/x)$$ and deduce that $$p_0 = a_n \prod_{j=1}^{n} \left(1 - \alpha_j X\right)$$

Let $P$ be a polynomial of degree $p$ written in factored form $P = a_p \prod_{i=1}^{d} (X - \lambda_i)^{m_i}$, where $\lambda_1, \ldots, \lambda_d$ are the distinct complex roots of $P$ and $m_1, \ldots, m_d$ their multiplicities.
Write in factored form the polynomial $X^p P\left(\frac{1}{X}\right)$ and prove that if $P$ is reciprocal then for every integer $i$, $1 \leq i \leq d$, $\lambda_i$ is nonzero and $\frac{1}{\lambda_i}$ is a root of $P$ with multiplicity $m_i$.

Let $P$ be a polynomial of degree $p$ written in factored form $P = a_p \prod_{i=1}^{d} (X - \lambda_i)^{m_i}$, where $\lambda_1, \ldots, \lambda_d$ are the distinct complex roots of $P$ and $m_1, \ldots, m_d$ their multiplicities. Write in factored form the polynomial $X^p P\left(\frac{1}{X}\right)$ and prove that if $P$ is reciprocal then for every integer $i$, $1 \leq i \leq d$, $\lambda_i$ is nonzero and $\frac{1}{\lambda_i}$ is a root of $P$ with multiplicity $m_i$.

Show that $p \wedge p_0 = 1$ if and only if $p$ has no stable root.

Let $Q$ be a polynomial of degree $p$. We say that $Q$ is antireciprocal if $$Q(X) = -X^p Q\left(\frac{1}{X}\right)$$ Show that if $Q$ is antireciprocal, 1 is a root of $Q$ and there exists a polynomial $P$ that is constant or reciprocal such that $Q = (X-1)P$.

Let $Q$ be a polynomial of degree $p$. We say that $Q$ is antireciprocal if $$Q(X) = -X^p Q\left(\frac{1}{X}\right)$$ Show that if $Q$ is antireciprocal, 1 is a root of $Q$ and that there exists a polynomial $P$ that is constant or reciprocal such that $Q = (X-1)P$.

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Justify that there exists $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.

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 $-1$.

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Let $h$ be the polynomial of degree $n$ defined by $h(X) = X p'$, where $p'$ is the derivative polynomial of $p$. We denote by $h_0$ and $(p')_0$ the reciprocal polynomials of $h$ and $p'$ respectively.
Show that $h = np - \lambda (p')_0$, then that $h_0 = \lambda(np - Xp')$.

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 $a$.
Deduce that $R$ is reciprocal or antireciprocal.

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 $a$. Deduce that $R$ is reciprocal or antireciprocal.

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Verify that $p'$ is split over $\mathbf{R}$ then show that $h \wedge h_0 = 1$ and deduce that $p'$ has no stable root.

For every integer $j \in \llbracket 1, n \rrbracket$, we denote by $f_j$ the polynomial $$f_j = a_n \prod_{k=j+1}^{n}\left(1 - \alpha_k X\right) \prod_{k=1}^{j-1}\left(X - \alpha_k\right)$$ with, according to standard conventions, $\prod_{k=n+1}^{n}(1-\alpha_k X) = \prod_{k=1}^{0}(X - \alpha_k) = 1$.
Show that if there exist two integers $i, k$ such that $1 \leq i < k \leq n$ and $\alpha_i \alpha_k = 1$, then $\alpha_i$ is a root of each polynomial $f_j$, where $j \in \llbracket 1, n \rrbracket$, and that the family $(f_1, \ldots, f_n)$ is linearly dependent.

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.

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 $p = \lambda p_0$.
For every real number $r > 0$, we denote by $F(r) = J(p(rX))$.
Show that $$\lim_{r \rightarrow 1^-} \pi\left(\frac{n}{2(r-1)} F(r)\right) = n - \sigma(p)$$

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 $p = \lambda p_0$.
We admit that the map defined on $S_n(\mathbf{R})$ with values in $\mathbf{R}^n$ which associates to a symmetric matrix the $n$-tuple of its real eigenvalues counted with their multiplicities, arranged in decreasing order, is continuous.
Deduce that $\sigma(p) = n - 1 - \pi(J(p'))$.