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
MathSciNet review:
2988714
Fulltext PDF
Abstract 
References 
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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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P. T. Bateman, R. M. Stemmler, Waring's problem for algebraic number fields and primes of the form , Illinois J. Math. 6 (1962), 142156. MR 0138616 (25:2059)
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L. Carlitz, Solution to problem 140 (proposed by I. Connell), Canad. Math. Bull. 11 (1968), 165169.
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M. Griffin, M. Krusemeyer, Matrices as sums of squares, Linear and Multilinear Algebra 5 (1977/78), no. 1, 3344. MR 0447170 (56:5485)
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S. A. Katre, S. A. Khule, A discriminant criterion for matrices over orders in algebraic numbers fields to be sums of squares, Proc. Symp. on Alg. and Number Th. (Cochin, 1990), 20, Centre Math. Sci., Trivandrum (1990), 3138. MR 1155423 (92m:11125)
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S. A. Katre, S. A. Khule, Matrices over orders in algebraic number fields as sums of th powers, Proc. Amer. Math. Soc. 128 (2000), no. 3, 671675. MR 1646194 (2000e:11026)
 6.
S. A. Katre, D. N. Sheth, matrices as sums of cubes, Preprint.
 7.
M. Newman, Sums of squares of matrices, Pacific J. Math. 118 (1985), no. 2, 497506. MR 789189 (86k:15011)
 8.
D. R. Richman, The Waring problem for matrices, Linear and Multilinear Algebra 22 (1987), no. 2, 171192. MR 936570 (89d:11087)
 9.
R. M. Stemmler, The easier Waring problem in algebraic number fields, Acta Arith. 6 (1960/1961), 447468. MR 0125834 (23:A3131)
 10.
L. N. Vaserstein, Every integral matrix is the sum of three squares, Linear and Multilinear Algebra 20 (1986), no. 1, 14. MR 875759 (88e:15009)
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L. N. Vaserstein, On the sum of powers of matrices, Linear and Multilinear Algebra 21 (1987), no. 3, 261270. MR 928280 (89a:15016)
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L. N. Vaserstein, Waring's problem for commutative rings, J. Number Theory 26 (1987), no. 3, 299307. MR 901242 (89e:11060)
 13.
L. N. Vaserstein, Waring's problem for algebras over fields, J. Number Theory 26 (1987), no. 3, 286298. MR 901241 (89e:11059)
 14.
Kshipra G. Wadikar, S. A. Katre, Matrices over a commutative ring with unity as sums of cubes, Proc. of Internat. Conf. on Emerging Trends in Math. and Comp. Appl., Dec. 1618, , Sivakasi, India, Allied Publ. (2010), 812.
 15.
KANT/KASH, Computational Algebraic Number Theory Software/ KAnt SHell, Version , http://www.math.tuberlin.de/kant/kash.html
 1.
 P. T. Bateman, R. M. Stemmler, Waring's problem for algebraic number fields and primes of the form , Illinois J. Math. 6 (1962), 142156. MR 0138616 (25:2059)
 2.
 L. Carlitz, Solution to problem 140 (proposed by I. Connell), Canad. Math. Bull. 11 (1968), 165169.
 3.
 M. Griffin, M. Krusemeyer, Matrices as sums of squares, Linear and Multilinear Algebra 5 (1977/78), no. 1, 3344. MR 0447170 (56:5485)
 4.
 S. A. Katre, S. A. Khule, A discriminant criterion for matrices over orders in algebraic numbers fields to be sums of squares, Proc. Symp. on Alg. and Number Th. (Cochin, 1990), 20, Centre Math. Sci., Trivandrum (1990), 3138. MR 1155423 (92m:11125)
 5.
 S. A. Katre, S. A. Khule, Matrices over orders in algebraic number fields as sums of th powers, Proc. Amer. Math. Soc. 128 (2000), no. 3, 671675. MR 1646194 (2000e:11026)
 6.
 S. A. Katre, D. N. Sheth, matrices as sums of cubes, Preprint.
 7.
 M. Newman, Sums of squares of matrices, Pacific J. Math. 118 (1985), no. 2, 497506. MR 789189 (86k:15011)
 8.
 D. R. Richman, The Waring problem for matrices, Linear and Multilinear Algebra 22 (1987), no. 2, 171192. MR 936570 (89d:11087)
 9.
 R. M. Stemmler, The easier Waring problem in algebraic number fields, Acta Arith. 6 (1960/1961), 447468. MR 0125834 (23:A3131)
 10.
 L. N. Vaserstein, Every integral matrix is the sum of three squares, Linear and Multilinear Algebra 20 (1986), no. 1, 14. MR 875759 (88e:15009)
 11.
 L. N. Vaserstein, On the sum of powers of matrices, Linear and Multilinear Algebra 21 (1987), no. 3, 261270. MR 928280 (89a:15016)
 12.
 L. N. Vaserstein, Waring's problem for commutative rings, J. Number Theory 26 (1987), no. 3, 299307. MR 901242 (89e:11060)
 13.
 L. N. Vaserstein, Waring's problem for algebras over fields, J. Number Theory 26 (1987), no. 3, 286298. MR 901241 (89e:11059)
 14.
 Kshipra G. Wadikar, S. A. Katre, Matrices over a commutative ring with unity as sums of cubes, Proc. of Internat. Conf. on Emerging Trends in Math. and Comp. Appl., Dec. 1618, , Sivakasi, India, Allied Publ. (2010), 812.
 15.
 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
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.
