A Six Generalized Squares Theorem, with Applications to Polynomial Identity Algebras

The theories of superalgebras and of P.I. algebras lead to a natural 2-graded extension of the integers. For these generalized integers, a “six generalized squares” theorem is proved, which can be considered as a 2-graded analogue of the classical “four squares” theorem for the natural numbers. This theorem was conjectured by A. Berele and A. Regev (“Exponential Growth of Some P.I. Algebras,” [[BR2]]) and has applications to p.i. algebras.