Phelps–Rodriguez Conjecture

Let $p$ be a complex polynomial of degree $n\ge2$ whose zeros all lie in the closed unit disk. For every zero $a$ of $p$, there is a critical point $\zeta$ satisfying $|\zeta-a|<1$, except when $|a|=1$ and $p$ is a nonzero scalar multiple of $z^n-a^n$.

Phelps-Rodriguez implies Sendov, so this entry records the stronger of the pair; the companion Sendov entry records the weaker statement and Mazur's original formalization, which proved Sendov but never stated the equality classification. The exceptional family is genuinely attained rather than an artefact of the proof: for p = z^n - 1 and a = 1 the only critical point is the origin, at distance exactly 1. Both conjectures fell out of one argument, and the strict form was not the announced target - Tao's digestion of Mazur's proof turned out to establish it, which is how a 1972 conjecture was resolved as a by-product of resolving a 1959 one.