On the Sendov-Ilyeff Conjecture

The Sendov Conjecture for Polynomials With at Most Seven Distinct Zeros

“…[7]). A similar theorem is proved in [6] as well, only the upper bound obtained there is somewhat weaker. 1 + mm ln(n -(n -l)a) o<a<i ln(l+a-a2) then I{p) < 1, unless n < 4 and p(z) = z" -e' ® for some real 6 ([..] denotes here the integral pari and the function between brackets is defined as n -1 for a = 1).…”

“…Various other special cases have been dealt with (see [3], [9], and [18] for references), the latest to date being that of polynomials with at most six different zeros, and more generally polynomials in P"(i/(n)), where v{n) is an increasing and unbounded function of n (Theorem 2.4 below) (cf. [2]; see also [6]). …”

“…These have been successful only for polynomials of degree n = 2,3,4,5, and more recently 6 (see [2] and [6] for this last case). It has also been proved that those roots of p lying sufEciently dose to the Unit circle satisfy an even stronger condition than the one stated in Sendov's conjecture ( [13], [19]).…”

“…I would like to thank Professor Arne Meurman for his help while working on this paper. Thanks are also due to E. S. Katsoprinakis for making his preprint article [6] available.…”

Proof of The Sendov Conjecture for Polynomials of Degree at Most Eight

“…1 We set R s and A s 0.562. Since p is extremal and G 1, we 9 9 j 2 assert that it must have the form < To verify this we set r s a y z and note that since z is repeated 0 0 Ž . r s r s r s , Re z G 0 and G 1, we must have…”