Small-span Hermitian matrices over quadratic integer rings

A totally real polynomial in Z[x] with zeros α1 6 α2 6 · · · 6 αn has span αn − α1. Building on the classification of all characteristic polynomials of integer symmetric matrices having span less than 4, we obtain a classification of polynomials having span less than 4 that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. We obtain as characteristic polynomials some (but not all) low-degree small-span polynomials that are not the characteristic (or minimal) polynomial of any integer symmetric matrix.