This paper deals with the asymptotic behavior of the sequence of codimensions c n cn(A), n = 1, 2,.., n=1,2,\ldots, of an algebra A over a field of characteristic zero. It is shown that when such sequence is polynomially bounded, then lim sup n → ∞ e log n e cn (A) and lim inf n → ∞ e log n e cn (A) cn(A) can be arbitrarily distant. Also, in case the codimensions are exponentially bounded, we can construct an algebra A such that exp e (A) = 2 \exp(A)=2 and, for any q ≥ 1 q≥ 1, there are infinitely many integers n such that cn (A) > nq 2n.This gives counterexamples to a conjecture of Regev for both cases of polynomial and exponential codimension growth.

Giambruno, A., Zaicev, M. (2016). Anomalies on codimension growth of algebras. FORUM MATHEMATICUM, 28(4), 649-656 [10.1515/forum-2014-0185].

### Anomalies on codimension growth of algebras

#### Abstract

This paper deals with the asymptotic behavior of the sequence of codimensions c n cn(A), n = 1, 2,.., n=1,2,\ldots, of an algebra A over a field of characteristic zero. It is shown that when such sequence is polynomially bounded, then lim sup n → ∞ e log n e cn (A) and lim inf n → ∞ e log n e cn (A) cn(A) can be arbitrarily distant. Also, in case the codimensions are exponentially bounded, we can construct an algebra A such that exp e (A) = 2 \exp(A)=2 and, for any q ≥ 1 q≥ 1, there are infinitely many integers n such that cn (A) > nq 2n.This gives counterexamples to a conjecture of Regev for both cases of polynomial and exponential codimension growth.
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Giambruno, A., Zaicev, M. (2016). Anomalies on codimension growth of algebras. FORUM MATHEMATICUM, 28(4), 649-656 [10.1515/forum-2014-0185].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10447/219017`
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