2003/12/05 by Michael Miller, Miller, Michael
Mathematics · #30C15 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30C15
paper · pdf · doi:10.48550/arxiv.math/0312130
5 pages, AMS-LaTeX, no figures. v2: proved Conjecture 1 for polynomials with all roots on a line, noted additional implications of Conjecture 1
arxiv created 2004/12/21 · arxiv updated 2009/12/01
A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if β is one of those roots, then within one unit of β lies a root of the polynomial's derivative. If we define r(β) to be the greatest possible distance between β and the closest root of the derivative, then Sendov's conjecture claims that r(β) ≤ 1. In this paper, we assume (without loss of generality) that 0 ≤ β≤ 1 and make the stronger conjecture that r(β) ≤ 1-(3/10)β(1-β). We prove this new conjecture for all polynomials of degree 2 or 3, for all real polynomials of degree 4, and for all polynomials of any degree as long as all their roots lie on a line or β is sufficiently close to 1.