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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 ${f: {\bf C} \rightarrow {\bf C}}$ be a polynomial of degree ${n \geq 2}$ that has all zeroes in the closed unit disk ${\{ z: |z| \leq 1 \}}$. If ${\lambda_0}$ is one of these zeroes, then ${f'}$ has at least one zero in ${\{z: |z-\lambda_0| \leq 1\}}$.

It is common in the literature on this problem to normalise ${f}$ to be monic, and to rotate the zero ${\lambda_0}$ to be an element ${a}$ of the unit interval ${[0,1]}$. As it turns out, the location of ${a}$ on this unit interval ${[0,1]}$ ends up playing an important role in the arguments.

Many cases of this conjecture are already known, for instance

• When ${n<9}$ (Brown-Xiang 1999);
• When ${a=0}$ (Gauss-Lucas theorem);
• When ${a \leq \frac{1}{n-1}}$ (Bojanov 2011);
• When ${c \leq a \leq 1-c}$ for a fixed ${c>0}$, and ${n}$ is sufficiently large depending on ${c}$ (Dégot 2014);
• When ${C n^{-1/7} \leq a \leq 1 - C n^{-1/4}}$ for a sufficiently large absolute constant ${C}$ (Chalebgwa 2020);
• When ${a=1}$ (Rubinstein 1968; Goodman-Rahman-Ratti 1969; Joyal 1969);
• When ${a \geq 1-\varepsilon_n}$, where ${\varepsilon_n>0}$ is sufficiently small depending on ${n}$ (Miller 1993; Vajaitu-Zaharescu 1993);
• When ${a \geq 1 - \frac{1}{2 n^9 4^n}}$ (Chijiwa 2011);
• When ${a \geq 1 - \frac{90}{n^{12} \log n}}$ (Kasmalkar 2014).

In particular, in high degrees the only cases left uncovered by prior results are when ${a}$ is close (but not too close) to ${0}$, or when ${a}$ is close (but not too close) to ${1}$; see Figure 1 of my paper.

Our main result covers the high degree case uniformly for all values of ${a \in [0,1]}$:

Theorem 2 There exists an absolute constant ${n_0}$ such that Sendov’s conjecture holds for all ${n \geq n_0}$.

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 ${n_0}$. I believe that the compactness arguments can be replaced with quantitative substitutes that provide an explicit ${n_0}$, but the value of ${n_0}$ produced is likely to be extremely large (certainly much larger than ${9}$).

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 ${a = o(1/\log n)}$, Sendov’s conjecture holds for sufficiently large ${n}$.

Theorem 4 (Sendov’s conjecture near the unit circle) Under the additional hypothesis ${1-o(1) \leq a \leq 1 - \varepsilon_0^n}$ for a fixed ${\varepsilon_0>0}$, Sendov’s conjecture holds for sufficiently large ${n}$.

We approach these theorems using the “compactness and contradiction” strategy, assuming that there is a sequence of counterexamples whose degrees ${n}$ going to infinity, using various compactness theorems to extract various asymptotic objects in the limit ${n \rightarrow \infty}$, 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 ${f}$ to Sendov’s conjecture, let ${\lambda}$ be a zero of ${f}$ (chosen uniformly at random among the ${n}$ zeroes of ${f}$, counting multiplicity), and let ${\zeta}$ similarly be a uniformly random zero of ${f'}$. We introduce the logarithmic potentials

$\displaystyle U_\lambda(z) := {\bf E} \log \frac{1}{|z-\lambda|}; \quad U_\zeta(z) := {\bf E} \log \frac{1}{|z-\zeta|}$

and the Stieltjes transforms

$\displaystyle s_\lambda(z) := {\bf E} \frac{1}{z-\lambda}; \quad s_\zeta(z) := {\bf E} \log \frac{1}{z-\zeta}.$

Standard calculations using the fundamental theorem of algebra yield the basic identities

$\displaystyle U_\lambda(z) = \frac{1}{n} \log \frac{1}{|f(z)|}; \quad U_\zeta(z) = \frac{1}{n-1} \log \frac{n}{|f'(z)|}$

and

$\displaystyle s_\lambda(z) = \frac{1}{n} \frac{f'(z)}{f(z)}; \quad s_\zeta(z) = \frac{1}{n-1} \frac{f''(z)}{f'(z)} \ \ \ \ \ (1)$

and in particular the random variables ${\lambda, \zeta}$ are linked to each other by the identity

$\displaystyle U_\lambda(z) - \frac{n-1}{n} U_\zeta(z) = \frac{1}{n} \log |s_\lambda(z)|. \ \ \ \ \ (2)$

On the other hand, the hypotheses of Sendov’s conjecture (and the Gauss-Lucas theorem) place ${\lambda,\zeta}$ inside the unit disk ${\{ z:|z| \leq 1\}}$. Applying Prokhorov’s theorem, and passing to a subsequence, one can then assume that the random variables ${\lambda,\zeta}$ converge in distribution to some limiting random variables ${\lambda^{(\infty)}, \zeta^{(\infty)}}$ (possibly defined on a different probability space than the original variables ${\lambda,\zeta}$), also living almost surely inside the unit disk. Standard potential theory then gives the convergence

$\displaystyle U_\lambda(z) \rightarrow U_{\lambda^{(\infty)}}(z); \quad U_\zeta(z) \rightarrow U_{\zeta^{(\infty)}}(z) \ \ \ \ \ (3)$

and

$\displaystyle s_\lambda(z) \rightarrow s_{\lambda^{(\infty)}}(z); \quad s_\zeta(z) \rightarrow s_{\zeta^{(\infty)}}(z) \ \ \ \ \ (4)$

at least in the local ${L^1}$ sense. Among other things, we then conclude from the identity (2) and some elementary inequalities that

$\displaystyle U_{\lambda^{(\infty)}}(z) = U_{\zeta^{(\infty)}}(z)$

for all ${|z|>1}$. 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 ${\lambda^{(\infty)}}$ and one originating from ${\zeta^{(\infty)}}$, then the location where these Brownian motions first exit the unit disk ${\{ z: |z| \leq 1 \}}$ 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 ${\lambda^{(\infty)}}$, ${\zeta^{(\infty)}}$ 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 ${a = o(1)}$, then ${\lambda^{(\infty)}, \zeta^{(\infty)}}$ almost surely lie in the semicircle ${\{ e^{i\theta}: \pi/2 \leq \theta \leq 3\pi/2\}}$ and have the same distribution.
• (ii) If ${a = 1-o(1)}$, then ${\lambda^{(\infty)}}$ is uniformly distributed on the circle ${\{ z: |z|=1\}}$, and ${\zeta^{(\infty)}}$ is almost surely zero.

In case (i) (and strengthening the hypothesis ${a=o(1)}$ to ${a=o(1/\log n)}$ to control some technical contributions of “outlier” zeroes of ${f}$), we can use this information about ${\lambda^{(\infty)}}$ and (4) to ensure that the normalised logarithmic derivative ${\frac{1}{n} \frac{f'}{f} = s_\lambda}$ 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 ${f}$ has a zero at ${a = o(1)}$ and that ${f'}$ has no zeroes near ${a}$. 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 ${f(z) = z^n - 1}$, where the zeroes ${\lambda}$ of ${f}$ are uniformly distributed amongst the ${n^{th}}$ roots of unity (including at ${a=1}$), and the zeroes of ${f'}$ are all located at the origin. In my paper I also discuss a variant of this construction, in which ${f'}$ has zeroes mostly near the origin, but also acquires a bounded number of zeroes at various locations ${\lambda_1+o(1),\dots,\lambda_m+o(1)}$ inside the unit disk. Specifically, we take

$\displaystyle f(z) := \left(z + \frac{c_2}{n}\right)^{n-m} P(z) - \left(a + \frac{c_2}{n}\right)^{n-m} P(a)$

where ${a = 1 - \frac{c_1}{n}}$ for some constants ${0 < c_1 < c_2}$ and

$\displaystyle P(z) := (z-\lambda_1) \dots (z-\lambda_m).$

By a perturbative analysis to locate the zeroes of ${f}$, one eventually would be able to arrive at a true counterexample to Sendov’s conjecture if these locations ${\lambda_1,\dots,\lambda_m}$ were in the open lune

$\displaystyle \{ \lambda: |\lambda| < 1 < |\lambda-1| \}$

and if one had the inequality

$\displaystyle c_2 - c_1 - c_2 \cos \theta + \sum_{j=1}^m \log \left|\frac{1 - \lambda_j}{e^{i\theta} - \lambda_j}\right| < 0 \ \ \ \ \ (5)$

for all ${0 \leq \theta \leq 2\pi}$. However, if one takes the mean of this inequality in ${\theta}$, one arrives at the inequality

$\displaystyle c_2 - c_1 + \sum_{j=1}^m \log |1 - \lambda_j| < 0$

which is incompatible with the hypotheses ${c_2 > c_1}$ and ${|\lambda_j-1| > 1}$. In order to extend this argument to more general polynomials ${f}$, we require a stability analysis of the endpoint equation

$\displaystyle c_2 - c_1 + c_2 \cos \theta + \sum_{j=1}^m \log \left|\frac{1 - \lambda_j}{e^{i\theta} - \lambda_j}\right| = 0 \ \ \ \ \ (6)$

where we now only assume the closed conditions ${c_2 \geq c_1}$ and ${|\lambda_j-1| \geq 1}$. The above discussion then places all the zeros ${\lambda_j}$ on the arc

$\displaystyle \{ \lambda: |\lambda| < 1 = |\lambda-1|\} \ \ \ \ \ (7)$

and if one also takes the second Fourier coefficient of (6) one also obtains the vanishing second moment

$\displaystyle \sum_{j=1}^m \lambda_j^2 = 0.$

These two conditions are incompatible with each other (except in the degenerate case when all the ${\lambda_j}$ vanish), because all the non-zero elements ${\lambda}$ of the arc (7) have argument in ${\pm [\pi/3,\pi/2]}$, so in particular their square ${\lambda^2}$ 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 ${f''/f'}$, which one then integrates and exponentiates to get good control on ${f'}$, and then on a second integration one gets enough information about ${f}$ 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 ${a}$ was extremely close to ${1}$ via a similar perturbative analysis; the main novelty is to control the error terms not in terms of the magnitude of the largest zero ${\zeta}$ of ${f'}$ (which is difficult to manage when ${n}$ gets large), but rather by the variance of those zeroes, which ends up being a more tractable expression to keep track of.

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