On the nilpotency degree of the algebra with identity x^n=0
  Denote by C_{n,d} the nilpotency degree of a relatively free algebra generated by d elements and satisfying the identity x^n=0. Under assumption that the characteristic p of the base field is greater than n/2, it is shown that C_{n,d}<n^{log_2(3d+2)+1} and C_{n,d}<4 2^{n/2} d. In particular, it is established that the nilpotency degree C_{n,d} has a polynomial growth in case the number of generators d is fixed and p > n/2. For p\neq2 the nilpotency degree C_{4,d} is described with deviation 4 for all d. As an application, a finite generating set for the algebra R^{GL(n)} of GL(n)-invariants of d matrices is established in terms of C_{n,d}. Several conjectures are formulated. 
Artem A. Lopatin
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