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Sendov's conjecture says that if all roots of a polynomial lie within the unit disk, then for every root, there exists a critical point at a distance at most one from the root. I read that Sendov conjecture has been proved for polynomials of degree up to 8.

I was wondering if for polynomials $p(z)$ of degree larger than 8, the conjecture can be broken down to the following two sub-problems. Does anyone know if any of these are already established or if there are any similar/refined results ?

  1. If there is a root $z_{0}$ of $p(z)$ whose distance is more than unity from all critical points, then another root cannot be closer than unity to $z_{0}$.

  2. If statement 1 is true, all other roots of the polynomial $p(z)$ lie outside the unit circle centered around $z_{0}$. In that case, maybe by tracking the zeros of the real and complex parts of $p(z)$ (which are well behaved for large modulus) and relating it to the zeros of the real and complex part of $p'(z)$ will show the existence of a critical point in the sector centered around zero whose endpoints the intersections of the consecutive $Re(p(z))=0$ contours with the unit circle.

To illustrate question 2, I am attaching a plot of the zeros of real (red) and imaginary (blue) part of a polynomial of order 10 with one isolated root. The yellow and cyan curves are the zeros of the derivative. The intersection of red and blue curves are the roots of the polynomial. Any red and blue curve behaves like a ray for large modulus and must pass through one root. This enforces the laminar flow like pattern inside the unit circle around $z_{0}$ for all but one pair of red and blue curve (which intersect at the isolated root). enter image description here?

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  • $\begingroup$ This is an excellent question - shame it didn't get enough attention. $\endgroup$ – nbubis Jan 11 '18 at 2:09

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