Spectrally arbitrary patterns over finite fields

An n x n zero-nonzero pattern A is spectrally arbitrary over a field F provided that for each monic polynomial r(x) is an element of F[x] of degree n, there exists a matrix A over F with zero-nonzero pattern A such that the characteristic polynomial p(A)(x) r(x). In this article, we investigate several classes of zero-nonzero patterns over finite fields and algebraic extensions of Q. We prove that there are no spectrally arbitrary patterns over F-2 and show that the full 2 x 2 pattern is spectrally arbitrary over F if and only if F contains at least five elements. We explore an n x n pattern with precisely 2n nonzero entries that is spectrally arbitrary over finite fields F-q with q >= n(n+1)/2 + 1, as well as Q. We also investigate an interesting 3 x 3 pattern for which the algebraic structure of the finite field rather than just the size of the field is a critical factor in determining whether or not it is spectrally arbitrary. This pattern turns out to be spectrally arbitrary over Q(root-3).