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Small-span Hermitian matrices over quadratic integer rings

Author: Gary Greaves
Journal: Math. Comp. 84 (2015), 409-424
MSC (2010): Primary 11C08, 15B36; Secondary 05C22, 05C50, 11C20, 15B33
Published electronically: May 19, 2014
MathSciNet review: 3266968
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Abstract: A totally-real polynomial in $ \mathbb{Z}[x]$ with zeros $ \alpha _1 \leqslant \alpha _2 \leqslant \dots \leqslant \alpha _d$ has span $ \alpha _d - \alpha _1$. 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.

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Additional Information

Gary Greaves
Affiliation: Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan

Received by editor(s): November 19, 2012
Received by editor(s) in revised form: April 18, 2013, and May 2, 2013
Published electronically: May 19, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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