  An $n\times n$ sign pattern $S$, which is a matrix with entries $0,+,-$, is
called spectrally arbitrary if any monic real polynomial of degree $n$ can be
realized as a characteristic polynomial of a matrix obtained by replacing the
non-zero elements of $S$ by numbers of the corresponding signs. A sign pattern
$S$ is said to be a superpattern of those matrices that can be obtained from
$S$ by replacing some of the non-zero entries by zeros. We develop a new
technique that allows us to prove spectral arbitrariness of sign patterns for
which the previously known "Nilpotent Jacobian" method does not work. Our
approach leads us to solutions of numerous open problems known in the
literature. In particular, we provide an example of a sign pattern $S$ and its
superpattern $S'$ such that $S$ is spectrally arbitrary but $S'$ is not,
disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den
Driessche.

Combinatorics