On a strong form of a conjecture of Boyle and Handelman

Abstract

Let ρr,m(x, λ ): = (x − λ) rm=0 � r+i−1 i � x m−i λ i . In this paper it is shown that if λ1 ,...,λ n are complex numbers such that λ1 = λ2 = ... = λr > 0a nd 0≤ � n=1 λ k ≤ nλ k ,fo r 1 ≤ k ≤ m := n − r ,t hen n � i=1 (λ − λi) ≤ ρr,m(λ, λ1), for all λ ≥ 6.75λ1. (∗) Moreover, if r ≥ m ,t hen (∗) holds for all λ ≥ λ1, while if r< m, but r is close to m ,a ndn is large, one can lower the constant of6 .75 in the inequality (∗). The inequality (∗) is inspired by, and related to, a conjecture ofBoyle and Handelman on the nonzero spectrum ofa nonnegative matrix.