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

Small-span Hermitian matrices over quadratic integer rings