Artem Lopatin
UNICAMP
Nagata-Higman Theorem: past, present and future.
Abstract: By the well-known Nagata-Higman Theorem, which at first was proved by Dubnov and Ivanov in 1943, the nilpotency degree C_{n,d} of the relatively free associative d-generated algebra with the identity x^n=0 is less than 2^n in the case of a field of characteristic zero or characteristic greater than n. Then Kuzmin (1975) and Razmyslov (1974) established that n(n+1)/2<= C_{n,d}<= n^2 in characteristic zero case. But the case of positive characteristic is not so well understood. We will present the resent results on C_{n,d} for the case of a finite field, which were obtained together with Ivan Chestakov.