grandes-ecoles 2022 Q14

grandes-ecoles · France · mines-ponts-maths2__psi Roots of polynomials Determine coefficients or parameters from root conditions

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:
$$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$

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?

Paper Questions

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