grandes-ecoles 2014 QIII.C.1

grandes-ecoles · France · centrale-maths2__mp Roots of polynomials Existence or counting of roots with specified properties

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P_\alpha)$ the set of complex polynomials that commute with $P_\alpha$ under composition.
Show that the only complex numbers $\alpha$ such that $\mathcal{C}(P_\alpha)$ contains a polynomial of degree three are 0 and $-2$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P_\alpha)$ the set of complex polynomials that commute with $P_\alpha$ under composition.

Show that the only complex numbers $\alpha$ such that $\mathcal{C}(P_\alpha)$ contains a polynomial of degree three are 0 and $-2$.

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