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Matrices over orders in algebraic number fields
as sums of $k$-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
DOI: https://doi.org/10.1090/S0002-9939-99-05206-5
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 $n \geq k \geq 2$ every integral $n \times n$ matrix is a sum of seven $k$-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 $n \times n$ matrix $ (n \geq k \geq 2)$ over an order of an algebraic number field to be a sum of (seven) $k$-th powers.


References [Enhancements On Off] (What's this?)

  • 1. M. Newman, Sums of squares of matrices, Pacific J. Math. 118 (1985), 497-506. MR 86k:15011
  • 2. D. R. Richman, The Waring problem for matrices, Linear and Multi. Alg. 22(1987), 171-192.MR 89d:11087
  • 3. L. N. Vaserstein, Every integral matrix is a sum of three squares, Linear and Multi. Alg. 20(1986), 1-4.MR 88e:15009
  • 4. L. N. Vaserstein, On the sum of powers of matrices, Linear and Multi. Alg. 21 (1987), 261-270. MR 89a:15016

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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

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