On the Sendov Conjecture for Polynomials with at Most Six Distinct Roots

Abstract

In this paper we prove that Sendov's conjecture is true for polynomials of degree Ž . n s 6 we even determine the so-called extremal polynomials in this case , as well as for polynomials with at most six different zeros. We then generalize this last Ž . Ž . result to polynomials of degree n with at most n distinct roots, where n is an increasing and unbounded function of n. ᮊ

The Sendov Conjecture for Polynomials With at Most Seven Distinct Zeros

“…Remark 2.5. Actually, the proof given in [2] shows only that if a is a zero of some p € P"(i/(ti)), then I{a) < 1 if |a| < 1; since v{n) < n if n > 5, it follows from Theorem 2.2 that the same is true if |a| = 1. It is easily checked that u{n) is an increasing and unbounded function of n, and numerical computations show e.g.…”

“…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]).…”

Borcea’s Variance Conjectures on the Critical Points of Polynomials

“…In spite of many ingenious ideas originating in attempts to solve Sendov's conjecture, only partial results are known today, see for instance the respective chapters in the monographs [39,47,51]. It was the young Julius Borcea who pushed the positive solution to this problem up to degree seven [7,8], and it was the mature Julius Borcea who outlined a series of extensions and modifications of Sendov's conjecture in the context of statistics of point charges in the plane. The authors of the present note had the privilege to work closely with Borcea on this topic, individually, and on two occasions in full formation (generously supported by the Mathematics Institute at the Banff Center in Canada, and by the American Institute of Mathematics in Palo Alto).…”

“…In the late 1960s soon after the publication of Hayman's book, a series of papers by several authors proved Sendov's conjecture for polynomials of degree three, four and five in quick succession. Sendov's conjecture for polynomials of degree six turned out to be significantly more difficult; its solution in Julius Borcea's first research paper appeared more than a quarter century later in 1996 [7]. In his next paper published in the same year, he proved that the conjecture is true for polynomials of degree seven [8].…”