I’ve just uploaded to the arXiv my paper “Sendov’s conjecture for sufficiently high degree polynomials“. This paper is a contribution to an old conjecture of Sendov on the zeroes of polynomials:

Conjecture 1 (Sendov’s conjecture)Let be a polynomial of degree that has all zeroes in the closed unit disk . If is one of these zeroes, then has at least one zero in .

It is common in the literature on this problem to normalise to be monic, and to rotate the zero to be an element of the unit interval . As it turns out, the location of on this unit interval ends up playing an important role in the arguments.

Many cases of this conjecture are already known, for instance

- When (Brown-Xiang 1999);
- When (Gauss-Lucas theorem);
- When (Bojanov 2011);
- When for a fixed , and is sufficiently large depending on (Dégot 2014);
- When for a sufficiently large absolute constant (Chalebgwa 2020);
- When (Rubinstein 1968; Goodman-Rahman-Ratti 1969; Joyal 1969);
- When , where is sufficiently small depending on (Miller 1993; Vajaitu-Zaharescu 1993);
- When (Chijiwa 2011);
- When (Kasmalkar 2014).

In particular, in high degrees the only cases left uncovered by prior results are when is close (but not too close) to , or when is close (but not too close) to ; see Figure 1 of my paper.

Our main result covers the high degree case uniformly for all values of :

Theorem 2There exists an absolute constant such that Sendov’s conjecture holds for all .

In principle, this reduces the verification of Sendov’s conjecture to a finite time computation, although our arguments use compactness methods and thus do not easily provide an explicit value of . I believe that the compactness arguments can be replaced with quantitative substitutes that provide an explicit , but the value of produced is likely to be extremely large (certainly much larger than ).

Because of the previous results (particularly those of Chalebgwa and Chijiwa), we will only need to establish the following two subcases of the above theorem:

Theorem 3 (Sendov’s conjecture near the origin)Under the additional hypothesis , Sendov’s conjecture holds for sufficiently large .

Theorem 4 (Sendov’s conjecture near the unit circle)Under the additional hypothesis for a fixed , Sendov’s conjecture holds for sufficiently large .

We approach these theorems using the “compactness and contradiction” strategy, assuming that there is a sequence of counterexamples whose degrees going to infinity, using various compactness theorems to extract various asymptotic objects in the limit , and somehow using these objects to derive a contradiction. There are many ways to effect such a strategy; we will use a formalism that I call “cheap nonstandard analysis” and which is common in the PDE literature, in which one repeatedly passes to subsequences as necessary whenever one invokes a compactness theorem to create a limit object. However, the particular choice of asymptotic formalism one selects is not of essential importance for the arguments.

I also found it useful to use the language of probability theory. Given a putative counterexample to Sendov’s conjecture, let be a zero of (chosen uniformly at random among the zeroes of , counting multiplicity), and let similarly be a uniformly random zero of . We introduce the *logarithmic potentials*

*Stieltjes transforms*Standard calculations using the fundamental theorem of algebra yield the basic identities and and in particular the random variables are linked to each other by the identity On the other hand, the hypotheses of Sendov’s conjecture (and the Gauss-Lucas theorem) place inside the unit disk . Applying Prokhorov’s theorem, and passing to a subsequence, one can then assume that the random variables converge in distribution to some limiting random variables (possibly defined on a different probability space than the original variables ), also living almost surely inside the unit disk. Standard potential theory then gives the convergence and at least in the local sense. Among other things, we then conclude from the identity (2) and some elementary inequalities that for all . This turns out to have an appealing interpretation in terms of Brownian motion: if one takes two Brownian motions in the complex plane, one originating from and one originating from , then the location where these Brownian motions first exit the unit disk will have the same distribution. (In our paper we actually replace Brownian motion with the closely related formalism of balayage.) This turns out to connect the random variables , quite closely to each other. In particular, with this observation and some additional arguments involving both the unique continuation property for harmonic functions and Grace’s theorem (discussed in this previous post), with the latter drawn from the prior work of Dégot, we can get very good control on these distributions:

Theorem 5

- (i) If , then almost surely lie in the semicircle and have the same distribution.
- (ii) If , then is uniformly distributed on the circle , and is almost surely zero.

In case (i) (and strengthening the hypothesis to to control some technical contributions of “outlier” zeroes of ), we can use this information about and (4) to ensure that the normalised logarithmic derivative has a non-negative winding number in a certain small (but not too small) circle around the origin, which by the argument principle is inconsistent with the hypothesis that has a zero at and that has no zeroes near . This is how we establish Theorem 3.

Case (ii) turns out to be more delicate. This is because there are a number of “near-counterexamples” to Sendov’s conjecture that are compatible with the hypotheses and conclusion of case (ii). The simplest such example is , where the zeroes of are uniformly distributed amongst the roots of unity (including at ), and the zeroes of are all located at the origin. In my paper I also discuss a variant of this construction, in which has zeroes mostly near the origin, but also acquires a bounded number of zeroes at various locations inside the unit disk. Specifically, we take

where for some constants and By a perturbative analysis to locate the zeroes of , one eventually would be able to arrive at a true counterexample to Sendov’s conjecture if these locations were in the open lune and if one had the inequality for all . However, if one takes the mean of this inequality in , one arrives at the inequality which is incompatible with the hypotheses and . In order to extend this argument to more general polynomials , we require a stability analysis of the endpoint equation where we now only assume the closed conditions and . The above discussion then places all the zeros on the arc and if one also takes the*second*Fourier coefficient of (6) one also obtains the vanishing second moment These two conditions are incompatible with each other (except in the degenerate case when all the vanish), because all the non-zero elements of the arc (7) have argument in , so in particular their square will have negative real part. It turns out that one can adapt this argument to the more general potential counterexamples to Sendov’s conjecture (in the form of Theorem 4). The starting point is to use (1), (4), and Theorem 5(ii) to obtain good control on , which one then integrates and exponentiates to get good control on , and then on a second integration one gets enough information about to pin down the location of its zeroes to high accuracy. The constraint that these zeroes lie inside the unit disk then gives an inequality resembling (5), and an adaptation of the above stability analysis is then enough to conclude. The arguments here are inspired by the previous arguments of Miller, which treated the case when was extremely close to via a similar perturbative analysis; the main novelty is to control the error terms not in terms of the magnitude of the largest zero of (which is difficult to manage when gets large), but rather by the variance of those zeroes, which ends up being a more tractable expression to keep track of.

## 38 comments

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8 December, 2020 at 7:34 pm

BlogPythonlooks like it ends with an extra phase “The conjecture is”

[Corrected, thanks – T.]8 December, 2020 at 7:37 pm

allenknutsonIt’s nice to see unsolved problems in the theory of polynomials in one complex variable!

I gather from the multitude of partial results that people care about this problem. Is there any obvious reason other than “it’s odd to have an unsolved problem in the theory of polynomials in one complex variable”?

8 December, 2020 at 9:04 pm

Terence TaoWell, the zeroes and critical points of a polynomial are both of interest in many applications, so I imagine it would be natural to ask what general relationships exist between the two sets of points, beyond what the classical Gauss-Lucas theorem (in the complex case) or Rolle’s theorem (in the real case) provides.

9 December, 2020 at 12:32 am

Oliver KnillThere is just also the elegance of the problem. Mathematicians at the time of Gauss would have liked it. It will be attractive also in 100 years, especially if not yet solved. It can also lead to beautiful and accessible problems in a single variable calculus course, like the following simple case which could be used in a lecture on Rolle’s theorem (as mentioned before in the comments)

“If f(x) is a real polynomial of degree n larger 1 with n roots in [-1,1], then there exists a critical point in distance 1 or less of any of the roots. Why does n=1 not work? Find a polynomial where the distance is 1.”

There is also just the beauty of what one can see when looking at polynomials, their roots and the critical points. Cases with a randomly chosen polynomial with 20, or 100 roots with red critical points http://www.math.harvard.edu/~knill/various/sendov indicates that often the critical points are quite close to the roots like moons to planets but that this often also fails and roots can cluster without critical points nearby.

As the random case has been mentioned in the post, it begs for the question what the distribution of the critical points is in the n to infinity limit, if the roots of the polynomial are uniformly distributed in the unit disc. Uniform again?

9 December, 2020 at 5:55 am

folkerttWell yes, but an interesting question may be what happens for rational functions, entire functions and/or those that satisfy some growth condition. or the Riemann zeta function for example, or functions that satisfy some functional equation. Someone could have thought along these lines. Interesting topic for speculation.

9 December, 2020 at 7:40 am

adityaguharoyAlthough we do have many reasons to analyze the zeroes and poles (the latter being more well understood) of the Riemann zeta function, but I think analyzing does not fall in this league.

9 December, 2020 at 9:46 am

Terence TaoThere is some literature on zeroes of derivatives of the Riemann zeta function, starting with a classical paper of Speiser that the Riemann Hypothesis holds if and only if all the non-real zeroes of lie on or to the right of the critical line. In general, as one keeps differentiating the zeta function, it is numerically observable that more and more of the zeroes start drifting to the right of the critical line and cluster in curves; see for instance these slides of Binder et al.. The situation is actually rather analogous to what happens to the zeroes of the Riemann xi function under the de Bruijn heat flow (with the notable difference that the higher derivatives do not obey a functional equation that makes the zeroes symmetric around the critical line). It is not widely expected that a closer study of these higher derivatives is likely to yield any substantive progress towards questions such as the Riemann hypothesis, but there are some interesting numerical phenomena here (though most likely arising primarily from general facts about derivatives of meromorphic functions, rather than having specifically a number theoretic origin).

9 December, 2020 at 8:53 am

Terence TaoIt is a result of Kabluchko (confirming a conjecture of Pemantle and Rivin) that the critical points of a large degree polynomial with zeroes sampled from a fixed distribution, are also asymptotically distributed with respect to the same distribution.

The phenomenon you observed numerically that critical points tend to be “paired” to nearby zeroes has begun to be rigorously explained, see for instance this recent paper by O’Rourke and Williams. Note that as there are zeroes and critical points, the pigeonhole principle will force one zero to be “unpaired”. Visually, numerical experiments suggest that this unpaired zero lurks near the center of the cluster of zeroes, and this seems to make sense in view of Bocher’s theorem and mean field approximation, but as far as I know there is not even a rigorous formulation of this phenomenon at present.

There is also some recent work on the dynamics of zeroes of repeated derivatives of such polynomials, such as this paper of Steinerberger and a followup work of Hoskins and Kabluchko. It is also connected to the story of fractional free convolution powers which I recently wrote a paper with Shylakhtenko about.

9 December, 2020 at 5:11 am

Rajan AnantharamanDear Friend Many thanks for this and others by Terence Tao that / who I admire tho sometimes not understood Best wishes Rajan

9 December, 2020 at 10:11 am

CraigI was wondering if there’s more progress to be made by looking at an individual zero of as follows: Take , with a monic polynomial with all zeros in the unit disk (!= a). Take large (), find a zero of in the unit disk centered on . Then turn into a continuous variable and look at what happens to as a function of .

We get

I’m not sure what you can do with that. , where ranges over the zeros of , so the RHS is a sum of single pole terms, while the LHS is the sum of double pole terms — can we show that must vanish as approaches the boundary of the disk? Can we show that always avoids other zeros of as varies so that we don’t have zero pair annihilation/creation?

9 December, 2020 at 10:44 am

AnonymousAre there similar conjectures on possible (algebraic geometrical) relationships between the set of zeros and the set of critical points for polynomials maps of several (real or complex) variables ?

9 December, 2020 at 2:07 pm

Terence TaoI don’t know of any explicit conjectures, but presumably they would involve the structure of the associated amoebas.

9 December, 2020 at 7:34 pm

AnonymousWhat an incredible paper!!!

For your perfect information, may I suggest to complete your preprint by recent results concerning the cases n = 9 and n = 10 and their related references:

1) Zaizhao Meng. Proof of the Sendov conjecture for polynomials of degree nine. ArXiv, 17 May 2018.

2) D.S. Bhattarai. A proof of the Sendov conjecture for polynomials of degree ten. International Journal of Scientific and Research Publications, 9 (8):182-191, August 2019.

Moreover the Ph.D thesis of T.P. Chalebgwa is devoted to this conjecture. In particular he has published an interesting article extending a previous result given by Dégot.

9 December, 2020 at 7:48 pm

AnonymousSpam:

https://terrytao.wordpress.com/2020/11/28/holomorphic-images-of-disks/#comment-606842

https://terrytao.wordpress.com/2020/11/28/holomorphic-images-of-disks/#comment-606847

9 December, 2020 at 7:45 pm

AnonymMore precisely:

T.P. Chalebgwa. Sendov’s conjecture : A note on a paper of Dégot. ArXiv:1804.09953, 26 April 2018.

10 December, 2020 at 11:24 am

AnonymousIn order to response to a previous post, there exists another famous similar conjecture, Kurepa’s conjecture.

Such a conjecture remains an open problem nowadays, despite the proof of Barsky & Benzeghou published in the JTNB which was dismissed by themselves in the same journal.

10 December, 2020 at 12:02 pm

AnonymousI would like for other interested people to check out the following paper by M.Sc Theophilus Agama submitted last year where he claims to prove Sendov’s conjecture as an application of his theory –

T. Agama. Expansivity theory and proof of Sendov’s conjecture. ArXiv: 1907.12825, 30 July 2019

10 December, 2020 at 5:55 pm

Alexandre EremenkoOn p. 14, 3d line after (2.9) “tales” must be “takes”.

[Thanks, this will be corrected in the next revision of the ms. -T]10 December, 2020 at 6:03 pm

CrisFollowing the two last posts, a well-known connex conjecture is the lonely runner problem.

For example this problem was studied by T. Agama in a recent preprint entitled ‘Distribution of boundary points of expansion and application to the lonely runner conjecture’ which was uploaded to the ArXiv on september 5, 2019.

11 December, 2020 at 11:21 am

AnonymousIt seems that theorem 2 is sufficient to imply Sendov’s conjecture by the following (perturbative) simple argument:

Let be a polynomial of degree and has all its zeros in the closed unit disk.

Define for a (perturbation) parameter

(1)

It is easy to verify that for sufficiently small , the polynomial has precisely bounded zeros (counting multiplicities) converging to the corresponding zeros of as , while the remaining zeros of are unbounded and converging (on the Riemann sphere) to as .

Similarly, for sufficiently small , has precisely bounded zeros (counting multiplicities) converging to the corresponding zeros of as , while the remaining zeros of are unbounded and converging (on the Riemann sphere) to as .

For any sufficiently small , there is a sufficiently small such that each zero (, say) of has a corresponding zero of such that

(2) for

while the remaining unbounded zeros of have sufficiently large magnitudes, for instance

(3) for

Similarly, this sufficiently small parameter can be chosen to satisfy also the following similar constraints for the zeros of and

(2') for

(3') for

Now, observe that for each , the zero of is (by (2)) -approximated by the zero of . Since we have

(4)

Which implies (via a scaled version of theorem 2 for ) the existence of a zero of such that

(5)

From (4) and (5) we see that

(6)

Hence by (3') it follows (for sufficiently small ) that

(7)

(i.e. is a bounded(!) critical point of )

By (2') (with condition (7) for the index ) we have

(8)

From (2), (5), (8) we have

(9)

(where the index j is dependent on the index i)

Since (9) holds for any positive , it proves Sendov’s conjecture.

11 December, 2020 at 1:56 pm

Terence TaoUnfortunately the scaled version of Theorem 2 would require all zeroes of to be under control, not just the first , so I don’t think one can easily deduce (5) from (4).

11 December, 2020 at 4:19 pm

AnonymousTo all the interested readers, the aforementioned reference in the present blog of Dr. TP Chalebgwa entitled “Sendov conjecture : A note on a paper of Dégot” was in fact published in Analysis Mathematica 46(3):447-463 Springer Verlag NY September 2020.

11 December, 2020 at 4:30 pm

ChrisIt seems effectively that Sendov’s conjecture has been proven.

What about Kurepa’s conjecture now?

12 December, 2020 at 12:32 am

MikkeWhy not ?

There are so many conjectures.

But if you want to become Fields medalist, as Alexander Grothendieck or Terence Tao for instance, you should demonstrate the ultimate conjecture i.e. the Riemann hypothesis (even if a proof has been proposed since several years in theoretical physics by Prof. German Robredo Sierra of the University of Madrid) …

12 December, 2020 at 12:54 am

AnonymousTo the best of my knowledge, the Riemann hypothesis has not been proven up to now despite intensive research works since many decades and a proposition coming from Ouganda but later dismissed by the maths community.

12 December, 2020 at 1:26 am

AndyI do totally agree with you.

There are still plenty of conjectures in various and heterogeneous fields of maths and theoretical physics.

But no doubt that the Graal is Riemann’s hypothesis, one of the remaining unsolved problems of the century in the sense of the Clay Mathematics Institute.

12 December, 2020 at 9:14 pm

AnonymousTechical redirection prblems

If we clic on ‘Dégot 2014’ we are redirected to the right place.

But if we clic on ‘work of Dégot’ we return to the first page of the present blog and not to the announced page.

[Corrected, thanks – T.]13 December, 2020 at 6:38 pm

DavidDear Professor,

Do you have already updated your preprint (and your blog) with the proofs of the Sendov conjecture for polynomials of degree 9 and 10 (see the post of December 9 at 7:34 on your blog for the two corresponding references) ?

Best regards

13 December, 2020 at 8:14 pm

Terence TaoNo, as I have not been able to confirm the veracity of their arguments (note that there is a history of multiple incorrect attempts at proving this conjecture). The degree 9 preprint, which is unpublished, relies on a further preprint of that author that was placed on the arXiv in 2013 but still remains unpublished to this day, while the degree 10 paper, while technically published in a journal, is not written to professional standards. In neither case do I see convincing evidence of a suitable level of peer review.

13 December, 2020 at 11:37 pm

VesselinFor a rational function other than a polynomial, may there be a similar kind of relation between the critical points and the divisor of zeros and poles?

Also Sendov’s conjecture and some of its partial progresses tended to have a parallel to this mean value conjecture of Smale’s fro “The fundamental theorem of algebra and complexity theory”: *If is a degree- complex polynomial normalized by and , is there a critical point , , with ?* (The supposedly extremal of value coming from the supposedly extremal polynomial .) As far as I am aware, despite of the partial cases solved, for general polynomials the lowest proved absolute constant upper bound is still the "4" coming out of Koebe's quarter theorem? Apologies for the naive question: any chance of your method bearing on that problem also?

19 December, 2020 at 10:33 pm

Terence TaoMy guess would be that rational functions would be significantly less “rigid” than polynomials (as one can sense from e.g., Runge’s theorem) and that not much can be said here. For instance even the Gauss-Lucas theorem does not seem to have any reasonable analogue for rational functions.

I looked briefly at applying compactness methods to Smale’s conjecture, but it seems there is a lot less compactness present than in the Sendov conjecture problem, where Prokhorov’s theorem can be used to good effect on the empirical distribution of zeroes or critical points. If one takes a sequence of appropriately normalised putative counterexamples to Smale’s conjecture that asymptotically saturates the Koebe bound then one seems to get a sequence of polynomials that converges locally uniformly to (a branch of) the inverse of the Koebe function, away from the slit associated to that function. So all the critical points of the polynomials will cluster near this slit, or near infinity (and one of the critical points has to converge to , to saturate the bound). But there doesn’t seem to be much mileage one can then extract from the limit objects – I think one can show that the majority of zeroes of the polynomial will go to infinity, but that’s about it. Probably some rather different methods (perhaps from conformal geometry?) are needed to make progress on this problem.

14 December, 2020 at 4:42 pm

sadboibruinour mans Terrence Tao does it again while I just failed my 32A final 😪

14 December, 2020 at 5:03 pm

AnonymousDear Profes5or,

Following one of your previous answer on this post, I never heard of the IJSRP journal in maths too, but its impact factor is of 6.64.

Furthermore have you already studied the Kurepa conjecture?

Best,

15 December, 2020 at 4:02 am

TimFinding some $n_0$ and getting it as low as possible sounds like a polymath-project.

16 December, 2020 at 9:54 am

AnonymousSince he used compactness many times, don’t expect it to be smaller than some exponential tower.

20 December, 2020 at 2:53 pm

Terence TaoThis thought occurred to me also. As commented elsewhere, the use of compactness right now will likely make the bounds quite poor if one tries to directly quantify the arguments in my paper. As one lower bound for how far one can get, to cover the middle range of (not too close to 0 or to 1) I use the results of Chalebgwa, which only apply for , so the first value of for which this result is non-vacuous is about . But probably the arguments I have that cover the range when is close to 0, or close (but not too close) to 1, will have even worse requirements on . [In particular, my arguments rely on unique continuation theorems; the usual quantitative analogue of such theorems are the Carleman inequalities, which have exponential weights in them, and if this is combined in any way with some sort of pigeonholing argument then tower-exponential type bounds would be expected. On the other hand, we also rely a lot on the logarithmic potential, which as its name suggests could potentially turn a bound of size into a bound of size in certain circumstances. So there may be some hope of keeping the total bounds somewhat civilised in nature.]

But perhaps once a “second generation” proof of the large n case comes out that relies less on compactness methods, a polymath project could be somewhat productive. It does resemble Polymath8 in that (a) the arguments are quite modular and one could work independently in different aspects of it (e.g., improving the analysis of Chalebgwa would be one obvious direction), and (b) there is a specific numerical “score” that one can use to measure progress which (as in Polymath8) would be a decreasing natural number, thus giving a natural termination point of the project when the number stops decreasing.

17 December, 2020 at 10:38 pm

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20 December, 2020 at 4:59 pm

VladDear Professor,

Maybe we have to check first the expansivity theory of Agama who claimed in a preprint uploaded to ArXiv last summer to have proved the Sendov-Ilieff conjecture.

Furthermore the approach followed by Meng in his preprints of 2013 and 2018 may be precised in the same way.

Last but not least, as commented before, the published paper of Bhattarai seems to be interesting too (even if its asian journal of publication is still unknown both in the maths and in the theoretical physics communities).

[Please do not sockpuppet to give the illusion of multiple independent commenters. -T]