Midpoints and critical points

Abstract

For a degree $5$ real polynomial with roots $x_1\le \cdots \le x_5$ and roots $\hat{I}{¾}_1\le \cdots \le \hat{I}{¾}_4$ of its derivative, we set $z_j:=(x_j+x_{j+1})/2$, $1\le j\le 4$. We prove that one cannot have at the same time ${\min }_{1\le j\le 3}(z_{j+1}-z_j)\ge {\min }_{1\le j\le 3}(\hat{I}{¾}_{j+1}-\hat{I}{¾}_j)$ and ${\max }_{1\le j\le 3}(z_{j+1}-z_j)\ge {\max }_{1\le j\le 3}(\hat{I}{¾}_{j+1}-\hat{I}{¾}_j)$. The result settles a general question about midpoints and critical points of hyperbolic polynomials.