A digestion of the proof of Sendov’s conjecture

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. 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). Using an AI agent, I have been able to formalize the entire argument in Lean, extended to by some minor modifications to the proof. This formalization is more streamlined than the original formalization (it has about 15,000 lines of code, compared with around 90,000 for the original proof). We now prove Conjecture 3. The cases have long been known but need to be treated separately; a short proof using the machinery developed here is provided at the end of the post. Suppose now that we have a counterexample for some , thus one can find a degree polynomial with zeroes for some and , in the closed unit disk, whose critical points all lie a distance at least from . We use notation here in the non-asymptotic sense, thus means that for some absolute constant (independent of ). We will also use the notation to denote a quantity that is bounded in magnitude by . To capture the fact that the critical points lie at a distance at least from , we write these critical points as for some (non-zero) in the closed unit disk. Example 4 If and , then are the non-trivial roots of unity, while the are all equal to . Strictly speaking this is not actually a counterexample to Conjecture 3, because is not strictly less than one; nevertheless this is an important motivating near-counterexample for the arguments below. Example 5 A generalization of the previous example was studied in Section 4 of my paper. Here one took where was an asymptotic parameter going to infinity, was a low-degree polynomial for some , and were constants. This polynomial has a zero at , critical points at , and additional critical points near . If all the critical points were at distance at least one from , one would have and while if all the zeroes were in the unit disk, the calculations in my paper showed that Here denotes a quantity that goes to zero as . If one ignores the errors, one can show that these conditions are only simultaneously feasible if and all the vanish, but the argument was somewhat subtle (I had to proceed by inspecting the second Fourier coefficient of (1)). This illustrates the fact that the regime is particularly delicate. We now have two sets of points in the closed unit disk: and . They “communicate” with each other through the polynomial and its first derivative , both of which can be expressed in terms of either set of points (as well as and ). Indeed, if we normalize to be monic, then we can factor in terms of the zeroes as and thus upon differentiating Here and in the sequel we adopt the convention of removing singularities when dealing with expressions that involve multiplication by both and , by cancelling such terms first in the event that . In a similar vein, can be factored and thus on integrating (and using ) It is convenient to rule out the easy case right away. In this case we see from (3), (4) that which is absurd since the first product has magnitude at most one, and the second product has magnitude at least one. Thus we can assume henceforth that . By inspecting or at various natural locations, we can thus obtain a number of identities relating the to the . We record the ones that we actually need here: Lemma 6 (Communication identities) Let denote the function (i) (Centroid identity) We have That is to say, the centroid of the zeroes equals the centroid of the critical values. (ii) (Polar identity) We have (iii) (First origin identity) We have (iv) (Second origin identity) We have (Again, we are using the convention of removing singularities to deal with the case where some of the vanish.) Proof: For (i), we inspect the behavior of as . From (2) we have and thus on differentiating term by term Meanwhile, from (4) we have Comparing coefficients, we obtain the claim. For (ii), we consider the expression . On the one hand, from (2), (3) one has (Note from hypothesis that cannot be a critical point, so the denominator is non-zero.) On the other hand, from (4), (5) one has Equating the two identities, we obtain (ii) after some algebra. For (iii), we evaluate . From (2) we have while from (5) we have Equating the two identities, we obtain (iii) after some algebra using (6). For (iv), we similarly evaluate . From (3) we have while from (4) one has Equating the two identities, we obtain (iv) after some algebra using (6). Remarkably, the polynomial will play no further role in the argument: the identities in (i)-(iv), together with the hypotheses that and lie in the closed unit disk, will be sufficient by themselves to obtain a contradiction. Example 7 Continuing the example in Example 4, in (i) both sides vanish. In (ii), both sides are equal to one. For (iii) and (iv), we have , with both sides of (iii) equal to one, and both sides of (iv) equal to zero. Remark 8 The centroid identity is extremely classical, going back to this 1948 paper of Popoviciu. The comparison of the polynomial at a location and at the polar inversion of that location across the closed unit disk is a familiar trick in the literature; see, e.g., Lemma 5 and Theorem 8 of Dégot. The specific form of the polar identity is implicit in the first part of Section 5 of Mazur’s AI-generated proof, while the origin identities are extracted from equation (6.3) of that proof. The first origin identity is also very close to Theorem 6 of Dégot, while the second origin identity is similar to some identities appearing in the proof of Lemma 6 of Dégot, as well as the work of Mir–Nazir–Wani and (in the case) Rubinstein. The work of Meir–Sharma and Mir–Nazir–Wani also contain several further identities relating the to the ; see in particular Lemma 15 below. Variants of (5) also appear in Proposition 10 of Miller. Remark 9 The first origin identity (9) is already strong enough to handle asymptotically all examples of the form in Example 5, except in the endpoint case where vanish and the are all . Indeed, as the are in the closed unit disk, (9) implies that On the other hand, routine calculations (omitted here) show that leading asymptotically to the constraint But all terms here are non-negative (since ), so this forces a contradiction unless (and hence also ) and the all vanish. As mentioned in Example 5, the most delicate regime occurs when . It is convenient to introduce the normalized version of , thus , and the case corresponds to . Informally, measures how close is to (at the scale of ). A key role in the argument will be played by the mean of the , particularly the real part . As the all lie in the unit disk, the mean does also, so that and On the other hand, in the example in Example 4, is equal to the extremal value of , and . In Example 5, we have (and ). It will be convenient to work with the quadratic polynomial with a particular emphasis on the value at : One should primarily think of as a measure of how close is to . Clearly we have for all (note that is strictly less than ). The arguments will revolve around the relationship between and . Specifically, we will establish the following two inequalities below the fold. The first inequality, which we call the “polar inequality”, comes in three forms: Proposition 10 (Polar inequality) (i) (Raw polar inequality) We have