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Sendov's conjecture for sufficiently high degree polynomials

Terence Tao

TL;DR

The paper proves that Sendov's conjecture holds for all sufficiently large degrees by a compactness-contradiction framework that analyzes asymptotic behavior of polynomial zeros. It builds a probabilistic-analytic model with random zeros $\lambda$ and $\zeta$, deriving limiting objects $\lambda^{(\infty)}$ and $\zeta^{(\infty)}$ via potential theory, Balayage, and Stieltjes transforms. It handles two endpoint regimes, $a^{(\infty)}=0$ and $a^{(\infty)}=1$, proving that both lead to consistent limiting behavior that cannot violate the conjecture, and then tackles the origin and unit-circle delicate regimes with a mix of argument-principle arguments and Taylor expansions to derive final contradictions. While the method is non-constructive and yields an existence of $n_0$ without an explicit value, it lays groundwork toward a decidability result for the conjecture at high degrees and clarifies the asymptotic structure of zeroes and critical points of high-degree polynomials in the unit disk context.

Abstract

Sendov's conjecture asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all of its zeroes in closed unit disk $\{ z: |z| \leq 1 \}$, then for each such zero $λ_0$ there is a zero of the derivative $f'$ in the closed unit disk $\{ z: |z-λ_0| \leq 1 \}$. This conjecture is known for $n < 9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov's conjecture holds for $n \geq n_0$. For $λ_0$ away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for $λ_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $λ_0$ is extremely close to the unit circle); and for $λ_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.

Sendov's conjecture for sufficiently high degree polynomials

TL;DR

The paper proves that Sendov's conjecture holds for all sufficiently large degrees by a compactness-contradiction framework that analyzes asymptotic behavior of polynomial zeros. It builds a probabilistic-analytic model with random zeros and , deriving limiting objects and via potential theory, Balayage, and Stieltjes transforms. It handles two endpoint regimes, and , proving that both lead to consistent limiting behavior that cannot violate the conjecture, and then tackles the origin and unit-circle delicate regimes with a mix of argument-principle arguments and Taylor expansions to derive final contradictions. While the method is non-constructive and yields an existence of without an explicit value, it lays groundwork toward a decidability result for the conjecture at high degrees and clarifies the asymptotic structure of zeroes and critical points of high-degree polynomials in the unit disk context.

Abstract

Sendov's conjecture asserts that if a complex polynomial of degree has all of its zeroes in closed unit disk , then for each such zero there is a zero of the derivative in the closed unit disk . This conjecture is known for , but only partial results are available for higher . We show that there exists a constant such that Sendov's conjecture holds for . For away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when is extremely close to the unit circle); and for near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.

Paper Structure

This paper contains 9 sections, 11 theorems, 182 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Asymptotic notation
  4. Control of limiting random variables
  5. Proof of (i)
  6. Proof of (ii)
  7. Getting a contradiction near the origin
  8. A motivational near-counterexample
  9. Getting a contradiction near the unit circle

Key Result

Theorem 1.2

Sendov's conjecture is true for all sufficiently large $n$. That is, there exists an absolute constant $n_0$ such that Sendov's conjecture holds for $n \geq n_0$.

Figures (4)

  • Figure 1: A schematic depiction (not drawn to scale) of the ranges of $a$ covered by current (in blue) and past (in black) partial results towards Theorem \ref{['main-contr']}. In particular, Theorem \ref{['main-contr']} follows by combining Theorems \ref{['contra-origin']}, \ref{['contra-circle']} with various combinations of the previous work in degot, chalebgwa, kasmalkar, chijiwa, vz, miller or Remarks \ref{['degot-time']}, \ref{['low-cover']}, \ref{['miller-time']}; for instance one can use the results in chalebgwa, chijiwa to cover all remaining cases.
  • Figure 2: Non-zero points on the arc $\overline{D(0,1)} \cap \partial D(1,1)$ have argument in $[\pi/3,\pi/2]$ or $[-\pi/2,-\pi/3]$. In particular, points on this arc subtend an angle of less than $\pi/4$ with the imaginary axis at the origin, so that their square lies in the left half-plane.
  • Figure 3: A Brownian motion originating from the compact set $K$ is fairly likely to exit the disk $\overline{D(0,R)}$ (enclosed here by the dashed circle) in the arc $A$; but a Brownian motion originating from the semicircle $C$ (or from the thin lune between the two circular arcs on the left) is far more likely to exit in the complement of $A$ instead. As a consequence, a pair $\lambda, \zeta$ of random variables in $\overline{D(0,1)}$ cannot have the same (resp. almost the same) balayage if $\zeta$ is almost surely in (resp. near) $C$ and $\lambda$ lies in $K$ with positive (resp. large) probability.
  • Figure 4: In this picture the dashed circle is $\partial D(0,1)$, and the compact set $S$ consists of the circular arc $\overline{D(0,1)} \cap \partial D(1,1)$ and the six points $T$ marked with crosses. Given any fixed point $z$ in the complement of $S$ and given any fixed $\varepsilon$ with $0 < \varepsilon < |z|$, one can find a contour $\gamma$ avoiding $S$ that starts at $z$ and ends at a point of magnitude $\varepsilon$, which is always moving towards the origin in the sense of \ref{['gamm']}. In the absence of the points $T$, one can easily achieve this by choosing $\gamma$ to be an arc on a suitable circle connecting $z$ and the origin (in most cases one can take the circle whose tangent at the origin is vertical); if this arc happens to pass through some of the points of $T$, one can easily perturb the arc to avoid them since these elements are isolated in $S$.

Theorems & Definitions (32)

  • Conjecture 1.1: Sendov's conjecture
  • Theorem 1.2: Sendov's conjecture for sufficiently high degree polynomials
  • Theorem 1.3: Main theorem, asymptotic contradiction form
  • Example 1.4: Near-counterexample near unit circle
  • Example 1.5: Near-counterexample near origin
  • Lemma 1.6: Basic relations between $\lambda,\zeta$ and $f$
  • proof
  • Remark 1.7
  • Remark 1.8
  • Example 1.9
  • ...and 22 more