On the nilpotency degree of the algebra with identity xn=0

AbstractDenote by Cn,d the nilpotency degree of a relatively free algebra generated by d elements and satisfying the identity xn=0. Under assumption that the characteristic p of the base field is greater than n/2, it is shown that Cn,d<nlog2(3d+2)+1 and Cn,d<4⋅2n2d. In particular, it is established that the nilpotency degree Cn,d has a polynomial growth in case the number of generators d is fixed and p>n2. For p≠2 the nilpotency degree C4,d is described with deviation 3 for all d. As an application, a finite generating set for the algebra RGL(n) of GL(n)-invariants of d matrices is established in terms of Cn,d. Several conjectures are formulated