Let $H$ be a $n\times n$ non-negative matrix. I am interested in matrices $H$ that satisfy the following condition: $\lambda_{\max}(H) \leq \operatorname{tr}(H)$.
Moreover, let $x,y \in \mathbb C^n$. What can I infer about bi-homogeneous polynomial $h(x, y) = x^\top H y$?
For example for $n = 2$ my condition is equivalent to $\det(H) > 0$ thus a polynomial $h(x, y)$ doesn't have zeroes in $\mathbb H_+ \times \mathbb H_-$. Can I get something similar for $n>2$?