NILPOTENCY INDEX OF NIL-ALGEBRA OF NIL-INDEX 3

Nagata and Higman proved that any nil-algebra of finite nilindex is nilpotent of finite index. The Nagata-Higman Theorem can be formulated in terms of T-ideals. TheT-ideal generated by $a^n$ for all $a{\in}A$ is also generated by the symmetric polynomials. The symmetric polynomials play an importmant role in analyzing nil-algebra. We construct the incidence matrix with the symmetric polynomials. Using this incidence matrix, we determine the nilpotency index of nil-algebra of nil-index 3.