grandes-ecoles 2018 Q13

grandes-ecoles · France · x-ens-maths2__mp Roots of polynomials Multiplicity and derivative analysis of roots

We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$, with all roots in $[-1,1]$).
Show that $R_{N}$ is the square of a polynomial: $R_{N}(X) = U_{N}(X)^{2}$ where $U_{N}(1) = 1$ and $U_{N}(-1) = \pm 1$. What can we say about the parity of $U_{N}$?

We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$, with all roots in $[-1,1]$).

Show that $R_{N}$ is the square of a polynomial: $R_{N}(X) = U_{N}(X)^{2}$ where $U_{N}(1) = 1$ and $U_{N}(-1) = \pm 1$. What can we say about the parity of $U_{N}$?

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