grandes-ecoles 2014 QIII.B.4

grandes-ecoles · France · centrale-maths2__mp Roots of polynomials Divisibility and minimal polynomial arguments

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 $\mathcal{C}(P)$ 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 $\mathcal{C}(T_2) = \{-1/2\} \cup \{T_n, n \in \mathbb{N}\}$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.C.1 QIII.C.2 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.C.4