Identities in the varieties generated by the algebras of upper triangular matrices

Consider the algebra $UT_s$ of upper triangular matrices of size $s$ over an arbitrary field. Petrogradsky proved that the exponent of an arbitrary subvariety in $\operatorname{var}(UT_s)$ 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.