On the location of critical points of polynomials

Given a polynomial $p$ of degree $n \geq 2$ and with at least two distinct roots let $Z(p) = \{z : p(z) = 0\}$. For a fixed root $\alpha \in Z(p)$ we define the quantities $\omega(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p)\setminus \{\alpha\} \bigr\}$and $\tau(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p')\setminus \{\alpha\} \bigr\}$. We also define $\omega(p)$ and $\tau(p)$ to be the corresponding minima of $\omega(p,\alpha)$ and $\tau(p,\alpha)$ as $\alpha$ runs over $Z(p)$. Our main results show that the ratios $\tau(p,\alpha)/\omega(p,\alpha)$ and $\tau(p)/\omega(p)$ are bounded above and below by constants that only depend on the degree of $p$. In particular, we prove that $(1/n)\omega(p)\leq\tau(p)\leq\bigl(1/2\sin(\pi/n)\bigr)\omega(p)$, for any polynomial of degree $n$.