Matrices over orders in algebraic number fields as sums of $k$-th powers
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- by S. A. Katre and Sangita A. Khule PDF
- Proc. Amer. Math. Soc. 128 (2000), 671-675 Request permission
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
- Morris Newman, Sums of squares of matrices, Pacific J. Math. 118 (1985), no. 2, 497–506. MR 789189
- David R. Richman, The Waring problem for matrices, Linear and Multilinear Algebra 22 (1987), no. 2, 171–192. MR 936570, DOI 10.1080/03081088708817831
- Leonid N. Vaserstein, Every integral matrix is the sum of three squares, Linear and Multilinear Algebra 20 (1986), no. 1, 1–4. MR 875759, DOI 10.1080/03081088608817738
- L. N. Vaserstein, On the sum of powers of matrices, Linear and Multilinear Algebra 21 (1987), no. 3, 261–270. MR 928280, DOI 10.1080/03081088708817800
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
- Received by editor(s): April 21, 1998
- Published electronically: July 6, 1999
- Communicated by: David E. Rohrlich
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1646194
Dedicated: Dedicated to the memory of David R. Richman