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Varieties with at most quadratic growth

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Abstract

Let V be a variety of non-necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions c n (V), n = 1, 2, …, and here we study varieties of polynomial growth. Recently in [16], for any real number α, 3 < α < 4, a variety V was constructed satisfying C 1 n α < c n (V) < C 2 n α, for some constants C 1, C 2. Motivated by this result here we try to classify all possible growth of varieties V such that c n (V) < C n α, with 0 < α < 2, for some constant C. We prove that if 0 < α < 1 then, for n large, c n (V) ≤ 1, whereas if V is a commutative variety and 1 < α < 2, then limn→∞ log n c n (V) = 1 or c n (V) ≤ 1 for n large enough.

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Authors and Affiliations

  1. Department of Algebra and Geometric Computations, Ulyanovsk State University, Ulyanovsk, 432970, Russia

    S. Mishchenko

  2. Dipartimento di Metodi e Modelli Matematici, Università di Palermo, 90133, Palermo, Italy

    A. Valenti

Authors
  1. S. Mishchenko
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  2. A. Valenti
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Corresponding author

Correspondence to S. Mishchenko.

Additional information

The first author was partially supported by RFBR grant 07-01-00080.

the second author was partially supported by MIUR of Italy.

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Mishchenko, S., Valenti, A. Varieties with at most quadratic growth. Isr. J. Math. 178, 209–228 (2010). https://doi.org/10.1007/s11856-010-0063-4

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  • DOI: https://doi.org/10.1007/s11856-010-0063-4

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