Abstract
Consider the algebra UTs of upper triangular matrices of size s over an arbitrary field. Petrogradsky proved that the exponent of an arbitrary subvariety in var(UTs) exists and is an integer. We strengthen the estimates for the growth of these varieties and provide equivalent conditions for finding these exponents. Kemer showed that in the case of a ground field of characteristic zero there exists no varieties of associative algebras with growth intermediate between polynomial and exponential. We prove that this property extends to the case of the fields of arbitrary characteristic distinct from 2.
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