Small-span Hermitian matrices over quadratic integer rings

TL;DR

This paper classifies small-span polynomials that are characteristic polynomials of Hermitian matrices over quadratic integer rings, extending previous classifications from integer symmetric matrices to a broader algebraic setting.

Contribution

It provides a new classification of small-span polynomials as characteristic polynomials of Hermitian matrices over quadratic integer rings, expanding the understanding beyond integer symmetric matrices.

Findings

Identifies small-span polynomials realizable as Hermitian matrix characteristic polynomials over quadratic integer rings.

Shows existence of low-degree small-span polynomials not realizable as integer symmetric matrix polynomials.

Extends classification results from integer symmetric matrices to Hermitian matrices over quadratic integer rings.

Abstract

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.