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Waring's problem for matrices over orders in algebraic number fields

A. S. Gadre·S. A. Katre
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Abstract

In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We prove a discriminant criterion for every 2 by 2 matrix over an order R to be sums of cubes and fourth powers over R. We also show that if q is a prime and n greater equal 2, then every n by n matrix over the ring of integers O, of a quadratic number field is a sum of q-th powers (of matrices) over O if and only if q is coprime to the discriminant of the quadratic number field.

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A. S. Gadre, S. A. Katre. 2007-02-15. Waring's problem for matrices over orders in algebraic number fields. https://arxiv.org/abs/math/0702445

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