grandes-ecoles 2018 Q11

grandes-ecoles · France · x-ens-maths2__mp Roots of polynomials Location and bounds on roots

We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
Using the results of questions 8, 9, and 10, conclude that $R_{N}$ has all its roots in the interval $[-1,1]$.

We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation
$$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$

Using the results of questions 8, 9, and 10, conclude that $R_{N}$ has all its roots in the interval $[-1,1]$.

Paper Questions

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