Spectrally arbitrary zero-nonzero patterns and field extensions

An n x n matrix pattern is said to be spectrally arbitrary over a field F provided for every monic polynomial p(t) of degree n, with coefficients from F, there exists a matrix with entries from F, in the given pattern, that has characteristic polynomial p(t). Let E subset of F subset of K be an extension of fields. It is natural to ask whether a pattern that is spectrally arbitrary over F must also be spectrally arbitrary over E or K. In this article it is shown that if F is dense in K and K is a complete metric space, then any spectrally arbitrary or relaxed spectrally arbitrary pattern over F is relaxed spectrally arbitrary over K. It is also established that if E is an algebraically closed subfield of a field F, then any spectrally arbitrary pattern over F is spectrally arbitrary over E. The 2n Conjecture and the Superpattern Conjecture are explored over fields other than the real numbers. In particular, examples are provided to show that the Superpattern Conjecture is false over the field with 3 elements. (c) 2017 Elsevier Inc. All rights reserved.