Classifying the Minimal Varieties of Polynomial Growth

Let $\mathcal{V}$ be a variety of associative algebras generated byan algebra with $1$ over a field of characteristic zero. Thispaper is devoted to the classification of the varieties$\mathcal{V}$ which are minimal of polynomial growth (i.e., theirsequence of codimensions growth like $n^k$ but any proper subvarietygrows like $n^t$ with $t<k$). These varieties are the buildingblocks of general varieties of polynomial growth.It turns out that for $k\le 4$ there are only a finite number ofvarieties of polynomial growth $n^k$, but for each $k > 4$, thenumber of minimal varieties is at least $|F|$, the cardinality ofthe base field and we give a recipe of how to construct them.