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.
In this paper, we prove the Sendov conjecture for polynomials of degree nine. We use a new idea to obtain new upper bound for the $σ-$sum to zeros of the polynomial.
Recently GM Sofi & SA Shabir [arXive: 1903.01850v2 [math.GM] 6 Mar 2019] made an attempt to prove the Sendov's conjecture. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.
Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result of this paper is a proof of Sendov conjecture when the polynomial $P$ has a degree higher than a fixed integer $N$. We will give estimates of its integer $N$ in terms of $|a|$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P'$) of a polynomial which would contradict Sendov conjecture.
A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $β$ is one of those roots, then within one unit of $β$ lies a root of the polynomial's derivative. If we define $r(β)$ to be the greatest possible distance between $β$ and the closest root of the derivative, then Sendov's conjecture claims that $r(β) \le 1$. In this paper, we assume (without loss of generality) that $0 \le β\le 1$ and make the stronger conjecture that $r(β) \le 1-(3/10)β(1-β)$. We prove this new conjecture for all polynomials of degree 2 or 3, for all real polynomials of degree 4, and …
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…
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.
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…
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…
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…
Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated in terms of matrix theory, mathematical statistics or potential theory. Quite a few links between classical works in the geometry of polynomials and recent advances in the location of spectra of small rank perturbations of structured matrices are established. A couple of simple examples provide natural and sometimes sharp bounds for the proposed conjectures…
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 …
Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ 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 linearly inextensible, they are not locally maximal for Sendov's conjecture.
