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

The theories of superalgebras and of P.I. algebras lead to a natural Z(2)-g raded extension of the integers. For these generalized integers, a "six gen eralized squares" theorem is proved, which can be considered as a Z(2)-grad ed analogue of the classical "four squares" theorem for the natural numbers . This theorem was conjectured by A. Berele and A. Regev ("Exponential Grow th of Some P.I. Algebras," [BR2]) and has applications to p.i. algebras. (C ) 2001 Academic Press.