On Sendov's conjecture about critical points of a polynomial
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The derivative of a polynomial $p(z)$ of degree $n$, with respect to point $\alpha$ is defined by $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$. Let $p(z)$ be a polynomial having all its zeros in the unit disk $|z| \leq 1$. The Sendov conjecture asserts that if all the zeros of a polynomial $p(z)$ lie in the closed unit disk, then there must be a zero of $p'(z)$ within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of $D_{\alpha}p(z)$ with respect to a specific zero of $p(z)$ and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.
Abstract
The derivative of a polynomial $p(z)$ of degree $n$, with respect to point $\alpha$ is defined by $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$. Let $p(z)$ be a polynomial having all its zeros in the unit disk $|z| \leq 1$. The Sendov conjecture asserts that if all the zeros of a polynomial $p(z)$ lie in the closed unit disk, then there must be a zero of $p'(z)$ within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of $D_{\alpha}p(z)$ with respect to a specific zero of $p(z)$ and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.
참고문헌 (7)
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- J. E. Brown, "Proof of Sendov conjecture for polynomials of degree at most eight", J. Math. Anal. Appl, 232, pp.272~292 (1999)
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- Q. I Rahman, "Analytic Theory of Polynomials" (2002), Oxford University Press
- G. Schmeisser, "Bemerkungen zu einer Vermutung von Ilief", Math. Z, 111, pp.121~125 (1969)
- Tereence Tao, "Sendov's Conjecture for su_ciently high degree polynomials", arXiv:2012.04125v1, ~ (2012)