A six generalized squares theorem, with applications to polynomial identity algebras (Q5942791)

The paper under review establishes a very nice number-theoretical result. Namely the authors prove the following theorem: Consider the set \(\mathbb{Z}\times\mathbb{Z}\) of the ordered pairs \((a,b)\) of integers, and define addition in the usual way. As for the multiplication, set \((a,b)(c,d)=(ac+bd,ad+bc)\). (A misprint in the paper gives \((ac+bd,ad+bd)\) for the last expression.) One can view this as the ring \(B=\mathbb{Z}[t]\), \(t^2=1\) by identifying \(t\) with \((0,1)\). Let \(P\) be the set of the pairs \((r^2,r^2)\) and \((r^2+s^2,2rs)\) for \(r\), \(s\) non-negative integers. The set \(P\) is the set of ``generalized'' squares in \(\mathbb{Z}\times\mathbb{Z}\). Then the authors prove that for \(r\geq s\) nonnegative integers, the pair \((r,s)\) is a sum of at most 6 generalized squares. The authors justify the consideration of such rings by showing that they fit very well into the description of several phenomena in the theory of PI algebras and especially their graded versions. Following their reasoning, the above ring is the natural super (or 2-graded) version of the ring of rational integers. (Also submitted to MR).