The growth of varieties generated by upper-triangular matrices algebras

It is shown that if the characteristic of the basic field does not equal two, then there exists no variety of associative algebras whose growth is intermediate between polynomial and exponential. Let $UT_s$ be the algebra of upper triangular matrices of dimension $s$ over an arbitrary field. V. M. Petrogradsky proved that the exponent of any subvariety of $\operatorname{var}(UTs)$ exists and is an integer number. In his paper the growth estimates for such varieties are strengthened.