Small-span Hermitian matrices over quadratic integer rings

Abstract

A totally-real polynomial in Z [ x ] \mathbb {Z}[x] with zeros α 1 ⩽ α 2 ⩽ ⋯ ⩽ α d \alpha _1 \leqslant \alpha _2 \leqslant \dots \leqslant \alpha _d has span α d − α 1 \alpha _d - \alpha _1 . Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4 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.