Polynomial growth in semigroup varieties

In 1989 M. Sapir posed the problem of describing all semigroup varieties where every finitely generated (f.g.) semigroup has polynomial growth. Here we find the solution of this problem for the case of an arbitrary nonperiodic semigroup variety defined by a system of identities over a finite set of variables. We also show that there exists an algorithm to decide whether or not the given finite system of homogeneous semigroup identities defines a variety where every f.g. semigroup has polynomial growth.