Matrices over orders in algebraic number fields

as sums of -th powers

Authors:
S. A. Katre and Sangita A. Khule

Journal:
Proc. Amer. Math. Soc. **128** (2000), 671-675

MSC (1991):
Primary 11P05, 11R04, 15A33; Secondary 11C20, 11E25, 15A24

Published electronically:
July 6, 1999

MathSciNet review:
1646194

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Abstract | References | Similar Articles | Additional Information

Abstract: David R. Richman proved that for every integral matrix is a sum of seven -th powers. In this paper, in light of a question proposed earlier by M. Newman for the ring of integers of an algebraic number field, we obtain a discriminant criterion for every matrix over an order of an algebraic number field to be a sum of (seven) -th powers.

**1.**Morris Newman,*Sums of squares of matrices*, Pacific J. Math.**118**(1985), no. 2, 497–506. MR**789189****2.**David R. Richman,*The Waring problem for matrices*, Linear and Multilinear Algebra**22**(1987), no. 2, 171–192. MR**936570**, 10.1080/03081088708817831**3.**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**4.**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

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Additional Information

**S. A. Katre**

Affiliation:
Department of Mathematics, University of Pune, Pune-411007, India

Email:
sakatre@math.unipune.ernet.in

**Sangita A. Khule**

Affiliation:
Department of Mathematics, University of Pune, Pune-411007, India

DOI:
https://doi.org/10.1090/S0002-9939-99-05206-5

Keywords:
Algebraic number fields,
order,
sums of powers,
discriminant,
matrices

Received by editor(s):
April 21, 1998

Published electronically:
July 6, 1999

Dedicated:
Dedicated to the memory of David R. Richman

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 1999
American Mathematical Society