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Analogue of Gauss-Lucas theorem for complex polynomials
Last modified: 2014-04-29
Abstract
Let $S(\phi)= \{z:\;|\arg(z)|\geq \phi\}$ be a sector on the complex plane $\CC$. If $\phi\geq \pi/2$, then $S(\phi)$ is a convex set and, according to the Gauss-Lucas theorem, if a polynomial $p(z)$ has all its zeros on $S(\phi)$, then the same is true for the zeros of all its derivatives. The main theorem in this lecture, called {\bf Sector theorem}, asserts that if the polynomial $p(z)$ is with real and non negative coefficients, then the same is true also for $\phi < \pi/2$, when the sector is not a convex set on the complex plane.
The Sector theorem is applied to prove stronger Rolle's theorem for complex polynomials.
The Sector theorem is applied to prove stronger Rolle's theorem for complex polynomials.