Building on the classification of all characteristic polynomials of integer
symmetric matrices having small span (span less than 4), we obtain a
classification of small-span polynomials that are the characteristic polynomial
of a Hermitian matrix over some quadratic integer ring. Taking quadratic
integer rings as our base, we obtain as characteristic polynomials some
low-degree small-span polynomials that are not the characteristic (or minimal)
polynomial of any integer symmetric matrix.Comment: 16 page