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

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 .

**1.**Paul T. Bateman and Rosemarie M. Stemmler,*Waring’s problem for algebraic number fields and primes of the form (𝑝^{𝑟}-1)/(𝑝^{𝑑}-1)*, Illinois J. Math.**6**(1962), 142–156. MR**0138616****2.**L. Carlitz, Solution to problem 140 (proposed by I. Connell), Canad. Math. Bull.**11**(1968), 165-169.**3.**Malcolm Griffin and Mark Krusemeyer,*Matrices as sums of squares*, Linear and Multilinear Algebra**5**(1977/78), no. 1, 33–44. MR**0447170****4.**S. A. Katre and Sangita A. Khule,*A discriminant criterion for matrices over orders in algebraic numbers fields to be sums of squares*, Proceedings of the Symposium on Algebra and Number Theory (Cochin, 1990), Publication, vol. 20, Centre Math. Sci., Trivandrum, 1990, pp. 31–38. MR**1155423****5.**S. A. Katre and Sangita A. Khule,*Matrices over orders in algebraic number fields as sums of 𝑘th powers*, Proc. Amer. Math. Soc.**128**(2000), no. 3, 671–675. MR**1646194**, 10.1090/S0002-9939-99-05206-5**6.**S. A. Katre, D. N. Sheth, matrices as sums of cubes, Preprint.**7.**Morris Newman,*Sums of squares of matrices*, Pacific J. Math.**118**(1985), no. 2, 497–506. MR**789189****8.**David R. Richman,*The Waring problem for matrices*, Linear and Multilinear Algebra**22**(1987), no. 2, 171–192. MR**936570**, 10.1080/03081088708817831**9.**Rosemarie M. Stemmler,*The easier Waring problem in algebraic number fields*, Acta Arith.**6**(1960/1961), 447–468. MR**0125834****10.**Leonid N. Vaserstein,*Every integral matrix is the sum of three squares*, Linear and Multilinear Algebra**20**(1986), no. 1, 1–4. MR**875759**, 10.1080/03081088608817738**11.**L. N. Vaserstein,*On the sum of powers of matrices*, Linear and Multilinear Algebra**21**(1987), no. 3, 261–270. MR**928280**, 10.1080/03081088708817800**12.**L. N. Vaserstein,*Waring’s problem for commutative rings*, J. Number Theory**26**(1987), no. 3, 299–307. MR**901242**, 10.1016/0022-314X(87)90086-2**13.**L. N. Vaserstein,*Waring’s problem for algebras over fields*, J. Number Theory**26**(1987), no. 3, 286–298. MR**901241**, 10.1016/0022-314X(87)90085-0**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. 16-18, , Sivakasi, India, Allied Publ. (2010), 8-12.**15.**KANT/KASH, Computational Algebraic Number Theory Software/ KAnt SHell, Version , http://www.math.tu-berlin.de/kant/kash.html

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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

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.