On the location of critical points of polynomials

Abstract:

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 $\omega (p,\alpha ) := \min \bigl \{|\alpha - v| : v \in Z(p)\setminus \{\alpha \} \bigr \}$ and $\tau (p,\alpha ) := \min \bigl \{|\alpha - v| : v \in Z(p’)\setminus \{\alpha \} \bigr \}$. 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 $n$.