GoKawiilThis 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. Then if is a zero of , there exists a critical point of with .
All three of these conjectures were established for (in a sequence of papers culminating in this paper of Brown and Xiang) and for sufficiently large (in a paper of myself, which in turn built upon several partial results in this setting). This left the case of intermediate to be settled. My arguments used some qualitative ingredients (most notably analytic continuation) and as such did not easily lend themselves to quantifying the threshold of above which the argument was valid.
Recently, Lech Mazur was able to use an AI tool to resolve Sendov’s conjecture for all , with the proof verified in Lean. However, the AI-generated proof was not human-digested to be in the form of a publication-ready preprint; and it has taken me several days (with heavy AI assistance) to perform such a digestion, to place the proof in proper context with previous literature and to simplify and streamline the argument to highlight the main ideas. (Note: the above chat log only represents a portion of the digestion work: the rest was performed with pen and paper, or using some further AI agents.) The same arguments also give a new proof of Rubinstein’s theorem, which I also give below the fold.
One consequence of this digestion is that the argument in fact demonstrates Conjecture 3, and thus resolves both the Sendov conjecture and the Phelps–Rodriguez conjecture in full generality.
The proof ends up being remarkably elementary. No complex analysis is used other than the fundamental theorem of algebra (and very basic facts about Möbius transformations); and the deepest inequality used as input is the Maclaurin inequality (and we only need a special case of that inequality which can be derived from the arithmetic mean-harmonic mean inequality and an induction argument).
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