4 problems

Let $S$ be an $n\times n$ zero-nonzero pattern that is spectrally arbitrary with respect to $\mathbb{Q}$. The $2n$-conjecture over the rationals. Then $S$ has at least $2n$ nonzero…

Let $\mathbb{F}\subset\mathbb{K}$ be an extension of fields, and let $S$ be a zero-nonzero pattern that is spectrally arbitrary over $\mathbb{F}$. Field-extension conjecture. Then…

A sign pattern is a matrix whose entries are $0$, $+$, or $-$. It is spectrally arbitrary if every monic real polynomial of degree $n$ is the characteristic polynomial of an…

Spectral arbitrariness conjecture. The patterns $\mathcal{T}_n$ are spectrally arbitrary for all $n\geq 2$, and hence are potentially nilpotent.