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RESEARCH PRODUCT

Anomalies on codimension growth of algebras

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Abstract This paper deals with the asymptotic behavior of the sequence of codimensions c n ⁢ ( A ) ${c_{n}(A)}$ , 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 → ∞ ⁡ log n ⁡ c n ⁢ ( A ) ${\limsup_{n\to\infty}\log_{n}c_{n}(A)}$ and lim inf n → ∞ ⁡ log n ⁡ c n ⁢ ( A ) ${\liminf_{n\to\infty}\log_{n}c_{n}(A)}$ can be arbitrarily distant. Also, in case the codimensions are exponentially bounded, we can construct an algebra A such that exp ⁡ ( A ) = 2 ${\exp(A)=2}$ and, for any q ≥ 1 ${q\geq 1}$ , there are infinitely many integers n such that c n ⁢ ( A ) > n q ⁢ 2 n ${c_{n}(A)>n^{q}2^{n}}$ . This gives counterexamples to a conjecture of Regev for both cases of polynomial and exponential codimension growth.