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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Matrices over commutative rings as sums of $ k$-th powers


Authors: S. A. Katre and Anuradha S. Garge
Journal: Proc. Amer. Math. Soc. 141 (2013), 103-113
MSC (2000): Primary 11R04, 11R11, 11R29, 15B33
Published electronically: May 14, 2012
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Abstract: In this paper, for $ n,k \ge 2,$ and $ R$ a commutative and associative ring with unity, we give necessary and sufficient trace conditions for an $ n \times n$ matrix over $ R$ to be a sum of $ k$-th powers of matrices over $ R$. We also prove that for $ n\ge m \ge 1$, if every $ m \times m$ matrix over $ R$ is a sum of $ k$-th powers of matrices over $ R$, then so is every $ n \times n$ matrix. As concrete examples, we prove a discriminant criterion for every $ n \times n$ matrix over an order $ R$ in an algebraic number field to be a sum of cubes and fourth powers of matrices over $ R$. We also show that if $ q$ is a prime and $ n \geq 2$, then every $ n \times n$ matrix over the ring $ {\mathcal O}$ of integers of a quadratic number field $ K$ is a sum of $ q$-th powers of matrices over $ {\mathcal O}$ if and only if $ q$ is coprime to the discriminant of $ K$.


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Additional Information

S. A. Katre
Affiliation: Department of Mathematics, University of Pune, Pune-411007, India
Email: sakatre@math.unipune.ac.in

Anuradha S. Garge
Affiliation: Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai-400098, India
Email: anuradha@cbs.ac.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11297-3
PII: S 0002-9939(2012)11297-3
Keywords: Algebraic number fields, discriminant, matrices, orders, trace, sums of powers, Waring’s problem.
Received by editor(s): August 13, 2010
Received by editor(s) in revised form: January 12, 2011, January 17, 2011, and June 10, 2011
Published electronically: May 14, 2012
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.