Proof of the Sendov conjecture for polynomials of degree at most eight
The Sendov conjecture asserts that: if \[ p(z)=\prod\limits_{k=1}^n (z-z_k),\quad | z_k| \leq 1, \quad k=1,2,\dots ,n \;n\geq 2, \] then each of the disks \(| z-z_k| \leq 1\), \(k=1,2,\dots ,n\) contains at least one zero of \(p' (z)\). The authors prove this conjecture for polynomials of degree \(n\leq 8,\) which is the best result known so far. The used method is nicely presented and is a very clear one (it depends on the estimates of the product of the moduli of the critical points of \(p' (z)\)). However, the applied technique will not work for arbitrary \(n.\) Nevertheless, it works for polynomials of arbitrary degree \(n,\) but with at most eight distinct zeros. The idea of the proof is the following: working with so-called extremal polynomials \[ p(z)=(z-a)\prod\limits_{k=1}^{n-1} (z-z_k),\quad 0<a<1, \] and using several lemmas (improved versions of the existing ones) concerning involved quantities as functions of \(a\), the desired result has been obtained after detailed analysis (pretty long calculations).
scientific article; zbMATH DE number 1286149