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 $(n+1)\times (n+1)$ nonnegative matrix whose nonzero eigenvalues are: $\lambda_0 \geq \vert\lambda_i\vert$, $i=1,\ldots,r$, $r \leq n$, then for all $x \geq \lambda_0$,

$\prod_{i=0}^{r} (x-\lambda_i) \leq x^{r+1}-\lambda_0^{r+1}.$

$(\ast)$

To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when $2(r+1)\geq (n+1)$, while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when $n\leq 4$ 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 $(\ast)$, from which it follows that the Boyle-Handelman conjecture is true. Actually, we do not start from the assumption that the $\lambda_i$'s are eigenvalues of a nonnegative matrix, but that $\lambda_1,\ldots, \lambda_{r+1}$ satisfy $\lambda_0\geq \vert\lambda_i\vert$, $i=1,\ldots, r$, and the trace conditions:

$\sum_{i=0}^{r} \lambda_i^k \geq 0,$   for all$k \geq 1.$

$(\ast\ast)$

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 $k=1,\ldots,r$. We further improve here on earlier results of the authors concerning this stronger form of the Boyle-Handelman conjecture.