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On the sum of powers of matrices. (English) Zbl 0637.10041
This paper gives three theorems about matrices as sums of squares, and one about kth powers: Theorem 1. Let R be a ring with 1 and $A\in M\sb n(R)$ with $n\ge 2$. Then A is a sum of squares in $M\sb n(R)$ iff tr(A) is a sum of squares mod 2R. Let P(R) denote the smallest s such that every sum of squares in R is a sum of s squares (or $P(R)=\infty$ if there is no such s). Then: Theorem 2. Let R be a ring with 1 such that $P(R/2R)=m<\infty$. Then $P(M\sb nR)\le 5+[(m+1)/2]$ for $n\ge 2$ and $P(M\sb nR)\le 7$ for $n\ge 2m+2.$ Theorem 3. Let R be commutative with 1 (so $P(R/2R)=m\le 1)$ and $n\ge 2$; then $P(M\sb nR)\le 6$, $P(M\sb nR)\le 5$ for n even, $P(M\sb 4R)\le 4$, $P(M\sb 3R)\le 4$ and $P(M\sb 2R)\le 3.$ Theorem 4. Given k there exists n(k) such that $n\ge n(k)$ implies every matrix in $M\sb n{\bbfZ}$ is a sum of 10 kth powers.
Reviewer: C.Small
##### MSC:
 11C20 Matrices, determinants (number theory) 11P05 Waring’s problem and variants 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15B36 Matrices of integers
##### Keywords:
matrix rings; sums of squares
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