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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.17689 |
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| _version_ | 1866929645385940992 |
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| author | Benanti, F. S. Valenti, A. |
| author_facet | Benanti, F. S. Valenti, A. |
| contents | Let $F$ be a field of characteristic zero and let $ \mathcal V$ be a variety of associative $F$-algebras. In \cite{regev2016} Regev introduced a numerical sequence measuring the growth of the proper central polynomials of a generating algebra of $ \mathcal V$. Such sequence $c_n^δ(\mathcal V), \, n \ge 1,$ is called the sequence of proper central polynomials of $ \mathcal V$ and in \cite{GZ2018}, \cite{GZ2019} the authors computed its exponential growth. This is an invariant of the variety. They also showed that $c_n^δ(\mathcal V)$ either grows exponentially or is polynomially bounded. The purpose of this paper is to characterize the varieties of associative algebras whose exponential growth of $c_n^δ(\mathcal V)$ is greater than two. As a consequence, we find a characterization of the varieties whose corresponding exponential growth is equal to two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of varieties of algebras of proper central exponent equal to two Benanti, F. S. Valenti, A. Rings and Algebras Let $F$ be a field of characteristic zero and let $ \mathcal V$ be a variety of associative $F$-algebras. In \cite{regev2016} Regev introduced a numerical sequence measuring the growth of the proper central polynomials of a generating algebra of $ \mathcal V$. Such sequence $c_n^δ(\mathcal V), \, n \ge 1,$ is called the sequence of proper central polynomials of $ \mathcal V$ and in \cite{GZ2018}, \cite{GZ2019} the authors computed its exponential growth. This is an invariant of the variety. They also showed that $c_n^δ(\mathcal V)$ either grows exponentially or is polynomially bounded. The purpose of this paper is to characterize the varieties of associative algebras whose exponential growth of $c_n^δ(\mathcal V)$ is greater than two. As a consequence, we find a characterization of the varieties whose corresponding exponential growth is equal to two. |
| title | A characterization of varieties of algebras of proper central exponent equal to two |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2412.17689 |