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# Classifying the Minimal Varieties of Polynomial Growth

Published:2013-05-15
Printed: Jun 2014
• Antonio Giambruno,
Dipartimento di Matematica e Informatica , Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy
• Daniela La Mattina,
Dipartimento di Matematica e Informatica , Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy
• Mikhail Zaicev,
Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, 119992 Moscow
 Format: LaTeX MathJax PDF

## Abstract

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
 MSC Classifications: 16R10 - $T$-ideals, identities, varieties of rings and algebras 16P90 - Growth rate, Gelfand-Kirillov dimension

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