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 $\sigma(p) = n - 1 - \pi(J(p'))$.
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 $\sigma(p) = n - 1 - \pi(J(p'))$.