ELA, Volume 9, pp. 138-149, August 2002, abstract.

On a Strong Form of a Conjecture of Boyle and Handelman

Assaf Goldberger and Michael Neumann

Let 

  \rho_{r,m}(x,\lambda) := (x-\lambda)^r 
         \sum_{i=0}^m (r+i-1\choose i) x^{m-i}\lambda^i.  

In this paper it is shown that if \lambda_1,...,\lambda_n 
are complex numbers such that 

             \lambda_1=\lambda_2=...=\lambda_r>0 
and 

       0 <= \sum_{i=1}^n \lambda_i^{k} <= n \lambda_1^k, 

for 1 <= k <= m:=n-r, then

\prod_{i=1}^n (\lambda-\lambda_i) <= \rho_{r,m}(\lambda,\lambda_1), 
      for all \lambda >= 6.75 \lambda_1  (*)

Moreover, if r>=m, then (*) holds for all \lambda >= \lambda_1, 
while if r<m, but r is close to m, and n is large, one can 
lower the constant of 6.75 in the inequality (*). The inequality 
(*) is inspired by, and related to, a conjecture of Boyle and 
Handelman on the nonzero spectrum of a nonnegative matrix.