Spectrally arbitrary star sign patterns

An n × n sign pattern Sn is spectrally arbitrary if, for any given real monic polynomial g(x) of degree n, there is a real matrix having sign pattern Sn and characteristic polynomial g(x). All n × n star sign patterns that are spectrally arbitrary, and all minimal such patterns, are characterized. This subsequently leads to an explicit characterization of all n × n star sign patterns that are potentially nilpotent. It is shown that any super-pattern of a spectrally arbitrary star sign pattern is also spectrally arbitrary.