A piece of contribution, perhaps:

In dimension at least 2, you can easily find situations where

is identically zero. For instance, in dimension 2, choose

satisfying the assumptions, and define

by exchanging the columns of

. Then the roots of their determinants are the same (hence negative), while

has two identical columns hence is singular for any

. But maybe this is not allowed since all complex numbers are roots of

...

You should specify what you mean by "monic" in your problem. There is no obvious definition, I think. In dimension 1, if you discard this hypothesis, the theorem becomes clearly false (

has a positive root).