Small-span Hermitian matrices over quadratic integer rings
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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.References
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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
- MR Author ID: 986306
- Email: grwgrvs@gmail.com
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 84 (2015), 409-424
- MSC (2010): Primary 11C08, 15B36; Secondary 05C22, 05C50, 11C20, 15B33
- DOI: https://doi.org/10.1090/S0025-5718-2014-02836-6
- MathSciNet review: 3266968