A proof of Sendov's conjecture

Sendov's Conjecture: A note on a paper of Dégot

Sendov's conjecture states that if all the zeroes of a complex polynomial $P(z)$ of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of $P(z)$. In a paper that appeared in 2014, Dégot proved that, for each $a\in (0,1)$, there exists an integer $N$ such that for any polynomial $P(z)$ with degree greater than $N$, if $P(a) = 0$ and all zeroes lie inside the unit disk, the disk $|z-a|\leq 1$ contains a critical point of $P(z)$. Based on this result, we derive an explicit formula $\mathcal{N}(a)$ for each $a \in (0,1)$ and, consequently obtai…

Proof of Sendov's conjecture

Sendov's conjecture, which was first introduced in the last 50s, asserts that if all the zeros of a polynomial $p$ lie in the closed unit disk then for each zero there must be a critical point of $p$ within unit distance. This paper confirms the conjecture.

与Sendov猜想相关的一个结果

A result related to the Sendov conjecture

本文研究Sendov猜想，探讨给定多项式阶数$n \geq 2$的零点$z_j$时，是否存在满足$p'(ζ) = 0$且$|z_j - ζ| \leq 1$的零点$ζ$，并给出了部分研究结果。

Unexpected local extrema for the Sendov conjecture

Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P) to be the maximum of the distances from each of the roots of a polynomial P to that root's nearest critical point. In this notation, Sendov's conjecture asserts that d(P)<=1 for every P in S(n). Define P in S(n) to be locally extremal if d(P)>=d(Q) for all nearby Q in S(n), and note that maximizing d(P) over all locally extremal polynomials P would settle the Sendov conjecture. Prior to now, the only polynomials known to be locally extremal were of the form P(z)=c(z^n+a) with |a|=1. In this pape…

超越Sendov猜想：二次Tang--Zhang不等式

Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality

本文证明Sendov猜想的一个定量加强版本——二次Tang--Zhang不等式，给出临界点距离平方和的精确下界及等号条件，并提供Lean 4形式化验证。

C*-代数的高斯-卢卡斯定理和C*-代数的塞诺夫猜想

C*-algebraic Gauss-Lucas Theorem and C*-algebraic Sendov's Conjecture

本文基于罗伯逊的结果，引入了在某些单单位可交换C*-代数上映射的微分概念，推导出C*-代数的高斯-卢卡斯定理，并提出C*-代数的塞诺夫猜想，验证了二次多项式的塞诺夫猜想。

Sendov猜想对每个次数 $n\ge 10^{200000}$ 成立

Sendov's conjecture holds for every degree $n\ge 10^{200000}$

本文通过有效化Tao的紧性方法，证明Sendov猜想对所有次数至少为10^{200000}的多项式成立，给出了显式常数。

寻求Sendov猜想的一个二次精细化

Seeking a quadratic refinement of Sendov's conjecture

本文提出Sendov猜想的一个二次精细化猜想，即存在常数c>0使得r(β)≤1-cβ(1-β)，并在多种多项式类中验证了该猜想，实验数据表明c≈0.233。

A quadratic approximation to the Sendov radius near the unit circle

Define $S(n,β)$ to be the set of complex polynomials of degree $n \ge 2$ with all roots in the unit disk and at least one root at $β$. For a polynomial $P$, define $|P|_β$ to be the distance between $β$ and the closest root of the derivative $P'$. Finally, define $r_n(β)=\sup \{|P|_β: P \in S(n,β) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(β) \le 1$. In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(β) \sim 1 + C_1 (1-|β|) + C_2 (1-|β|)^2$ for $|β|$ near 1. We also conside…

Analyticity domains of critical points of polynomials. A proof of Sendov's conjecture

Let $¶_{n}^c(\barμ,\barν)$ be the set of all complex polynomials $p(z)=\prod_{i=1}^{m}(z-z_i)^{μ_i}$, $\sum_{i=1}^mμ_i=n$, with derivatives of the form $$ p'(z)=n\prod_{i=1}^{m}(z-z_i)^{μ_i-1}\prod_{j=1}^{k}(z-ξ_j)^{ν_j}, ~\sum_{j=1}^kν_j=m-1. $$ In this note we prove the following:\par\medskip {\it \noindent For a fixed ordering $α=(1,2,\ldots,m)$, the distinct zeros $\{z_i\}_{i=1}^m$ and the distinct critical points of the second kind $\{ξ_j\}_{j=1}^k$ of polynomials from $¶_{n}^c(\barμ,\barν)$ are analytic functions $\{z_i^{αβ}\}_{i=1}^m$ and $\{ξ_j^{αβ}\}_{j=1}^k$, resp., $β=(i_1,i_2,\ldot…

借助简化计算机辅助的森多夫猜想的标量优化证明

A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance

本研究完成森多夫猜想的标量优化证明，通过解析处理连续优化与无界次数参数，仅保留少量显式单变量不等式，大幅降低了所需的计算机辅助。

The critical points of a polynomial

In this paper, we obtain new results on the critical points of a polynomial, these results are useful to the Sendov conjecture.

Maximal and inextensible polynomials and the geometry of the spectra of normal operators

We consider the set S(n,0) of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ we let $|p|_{0}$ denote the distance from the origin to the zero set of $p'$. We determine all 0-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are inextensible, they are not necessarily locally maximal for Sendov's conjecture. This invalidates the recently claimed …

博尔恰二次方差猜想的一个证明

A proof of Borcea's quadratic variance conjecture

该论文证明了博尔恰二次方差猜想，通过多项式积分恒等式和定向整数算术，验证了不同次数复多项式的相关零点与临界点距离结论，还给出森多夫猜想的新证明。

A proof of Smale's mean value conjecture

A proof of Smale's mean value conjecture from 1981 is given.

A self-contained proof to Martio's conjecture in the class of BLD-maps

We provide a self-contained proof to so-called Martio's conjecture in the class of mappings of bounded length distortion. Unlike the earlier proofs, our proof is not based on the modulus of continuity estimate of Martio from 1970.

Herbert Stahl's proof of the BMV conjecture

The paper contains a simplified version of Stahl's proof of a conjecture of Bessis, Moussa and Villani on the trace of matrices A+tB with Hermitean A and B.

New Proofs for the Abhyankar-Gurjar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture

We first give a new proof and also a new formulation for the Abhyankar-Gurjar inversion formula for formal maps of affine spaces. We then use the reformulated Abhyankar-Gurjar formula to give a more straightforward proof for the equivalence of the Jacobian Conjecture with a special case of the Vanishing Conjecture of (homogeneous) quadratic differential operators with constant coefficients.