Further strengthening of Rolle's theorem for complex polynomials (Q2216651)

Let \(D[c,r]\subset{\mathbb C}\) denote the closed disk with center \(c\) and radius \(r\). A domain \(\Theta_n\subset{\mathbb C}\) is called a Rolle domain if every complex polynomial \(p\) of degree \(n\) with \(p(\operatorname{i})=p(-\operatorname{i})\) has at least one critical point in it. Examples for Rolle domains are \(\Theta_n=D[0,\cot(\tfrac{\pi}{n})]\) or \(\Theta_n=D[-c,r]\cup D[c,r]\) with \(c=\cot(\tfrac{\pi}{n-1})\) and \(r=1/\sin(\tfrac{\pi}{n-1})\). In this paper, the authors find a Rolle domain of the form \(\Theta_n=D[-d,\rho]\cup D[d,\rho]\), where \(d\) and \(\rho\) are given rather explicitly. This domain has the smallest area of all Rolle domains consisting of two disks which are symmetric with respect to both axes.