Matrices over commutative rings as sums of th powers
Authors:
S. A. Katre and Anuradha S. Garge
Journal:
Proc. Amer. Math. Soc. 141 (2013), 103113
MSC (2000):
Primary 11R04, 11R11, 11R29, 15B33
Published electronically:
May 14, 2012
Fulltext PDF
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Additional Information
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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 D. R. Richman, The Waring problem for matrices, Linear and Multilinear Algebra 22 (1987), no. 2, 171192. MR 936570 (89d:11087)
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 L. N. Vaserstein, Waring's problem for algebras over fields, J. Number Theory 26 (1987), no. 3, 286298. MR 901241 (89e:11059)
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 KANT/KASH, Computational Algebraic Number Theory Software/ KAnt SHell, Version , http://www.math.tuberlin.de/kant/kash.html
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Additional Information
S. A. Katre
Affiliation:
Department of Mathematics, University of Pune, Pune411007, India
Email:
sakatre@math.unipune.ac.in
Anuradha S. Garge
Affiliation:
Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai400098, India
Email:
anuradha@cbs.ac.in
DOI:
http://dx.doi.org/10.1090/S000299392012112973
PII:
S 00029939(2012)112973
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 HuisgenZimmermann
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
