For a class \({\mathcal C}\) of algebras, and a cardinal \(k\), let \(G_{\mathcal C}(k)\) denote the number of pairwize non-isomorphic members of \({\mathcal C}\) that are generated by at most \(k\) elements. The function \(G_{\mathcal C}(k)\), restricted to positive integral \(k\), is called the growth-spectrum of \({\mathcal C}\). The authors say that a variety \({\mathcal V}\) has polynomially many models iff the growth-spectrum \(G_{\mathcal C}(k)\) is bounded above by some polynomial function.
A ring \(R\) is said to be of finite representation type if there are only finitely many finitely generated and directly indecomposible \(R\)-modules, up to isomorphism.
The main result is the following.
Theorem. A locally finite variety which omits type 1 of tame congruence theory (defined by D. Hobby and R. McKenzie in their book: The structure of finite algebra [Contemp. Math. 76, Am. Math. Soc., Providence, RI (1988; Zbl 0721.08001)]) has polynomially many models iff it is congruence-modular and affine over a finite ring of finite representation type.
The assumption “a variety omits type 1” is somewhat technical, but also rather weak, as it holds for all varieties of groups, rings, modules, lattices and for most of the varieties studied in algebraic logic. The authors believe that it will eventually prove possible to remove this assumption and obtain a full characterization.
In the conclusion, the authors state some open problems, show that for an affine variety \({\mathcal A}\) the function \(G_{\mathcal A}(k)\) is at most exponential – bounded by the function \(2^{ck^2}\) for some positive constant \(c\) – and present an example of an affine variety whose growth-spectrum is also bounded below by a function \(2^{ck^2}\), \(c>0\).