Matrices over commutative rings as sums of -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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11297-3

Published electronically:
May 14, 2012

MathSciNet review:
2988714

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Abstract: In this paper, for and a commutative and associative ring with unity, we give necessary and sufficient trace conditions for an matrix over to be a sum of -th powers of matrices over . We also prove that for , if every matrix over is a sum of -th powers of matrices over , then so is every matrix. As concrete examples, we prove a discriminant criterion for every matrix over an order in an algebraic number field to be a sum of cubes and fourth powers of matrices over . We also show that if is a prime and , then every matrix over the ring of integers of a *quadratic* number field is a sum of -th powers of matrices over if and only if is coprime to the discriminant of .

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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:
https://doi.org/10.1090/S0002-9939-2012-11297-3

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.