This post concerns the following conjecture of Sendov , as well as its strengthening by Phelps–Rodriguez :

Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with .

Conjecture 2 (Phelps–Rodriguez conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with , unless is on the unit circle and is a scalar multiple of .

By applying a rotation around the origin, we can normalize to be a real number with .

From the work of Rubinstein , both conjectures were already established in the case, so one can restrict to the case. Both of these conjectures then follow from

Conjecture 3 (Sendov’s conjecture in interior) Let . Let be a degree polynomial with all zeroes in the unit disk.…