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

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 o...

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Bibliographic Details
Published inarXiv.org
Main Authors Gadre, A S, Katre, S A
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.02.2007
Subjects
Cubes
Integers
Number theory
Rings (mathematics)
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Summary: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.
ISSN:2331-8422