Stronger Rolle’s Theorem for Complex Polynomials

Abstract:

The following Rolle’s Theorem for complex polynomials is proved. If $p(z)$ is a complex polynomial of degree $n\geq 5$, satisfying $p(-i)=p(i)$, then there is at least one critical point of $p$ in the union $D[-c;r] \cup D[c;r]$ of two closed disks with centres $-c, c$ and radius $r$, where \begin{equation*} c= \cot (2\pi /n),\;\;\; r=1/ \sin (2\pi /n). \end{equation*} If $n=3$, then the closed disk $D[0; 1/\sqrt {3}]$ has this property; and if $n=4$, then the union of the closed disks $D[-1/3; 2/3] \cup D[1/3; 2/3]$ has this property. In the last two cases, the domains are minimal, with respect to inclusion, having this property.

This theorem is stronger than any other known Rolle’s Theorem for complex polynomials.