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1. CJM 2013 (vol 66 pp. 625)

Giambruno, Antonio; Mattina, Daniela La; Zaicev, Mikhail
Classifying the Minimal Varieties of Polynomial Growth
Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with $1$ over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ which are minimal of polynomial growth (i.e., their sequence of codimensions growth like $n^k$ but any proper subvariety grows like $n^t$ with $t\lt k$). These varieties are the building blocks of general varieties of polynomial growth. It turns out that for $k\le 4$ there are only a finite number of varieties of polynomial growth $n^k$, but for each $k \gt 4$, the number of minimal varieties is at least $|F|$, the cardinality of the base field and we give a recipe of how to construct them.

Keywords:T-ideal, polynomial identity, codimension, polynomial growth,
Categories:16R10, 16P90

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