An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle Handelman conjecture

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A is an (n+1) ×(n+1) nonnegative matrix whose nonzero eigenvalues are: λ₀ ≥ |λ_i|, i = 1,..., r, r ≤ n, then for all x ≥ λo, (^*) Π^r _i=0(x-λ_i) ≤ x^r+1- λ^r+1₀. To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2(r + 1) ≥ (n+1), while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 4 and when the spectrum of A is real. They also showed that the conjecture is asymptotically true with the dimension. Here we prove a slightly stronger inequality than in (^*), from which it follows that the Boyle-Handelman conjecture is true. Actually, we do not start from the assumption that the λ_i's are eigenvalues of a nonnegative matrix, but that λ₁,..., λ_r+1 satisfy λ₀ ≥ |λ_i|, i = 1,..., r, and the trace conditions: (^**) Σ^r _i=0λ^k_i ≥ 0, for all k ≥ 1. A strong form of the Boyle-Handelman conjecture, conjectured in 2002 by the present authors, says that (^*) continues to hold if the trace inequalities in (^**) hold only for k = 1,..., r. We further improve here on earlier results of the authors concerning this stronger form of the Boyle-Handelman conjecture.