grandes-ecoles 2021 Q11

grandes-ecoles · France · centrale-maths2__official Roots of polynomials Proof of polynomial identity or inequality involving roots

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Show that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1 - \omega_k)^2} + nP(\lambda)$$ One may apply equality (I.2) to the polynomial $X^{2n}$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.

Show that
$$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1 - \omega_k)^2} + nP(\lambda)$$
One may apply equality (I.2) to the polynomial $X^{2n}$.

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