Polynomial codimension growth and the Specht problem

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• 作者：Antonio Giambruno^a ; ¹ ; ^{antonio.giambruno@unipa.it" class="auth_mail" title="E-mail the corresponding author} ; Sergey Mishchenko^b ; ^{mishchenkosp@mail.ru" class="auth_mail" title="E-mail the corresponding author} ; Angela Valenti^c ; ¹ ; ^{angela.valenti@unipa.it" class="auth_mail" title="E-mail the corresponding author} ; Mikhail Zaicev^d ; ² ; ^{zaicevmv@mail.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 17A30 ; 16R10 ; secondary ; 16P90
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：421-436
• 全文大小：366 K

文摘

We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α∈(0,1)$\alpha \in (0,1)$ a commutative nonassociative algebra A_α$A_{\alpha }$ whose codimension sequence c_n(A_α)$c_n(A_{\alpha })$, n=1,2,…$n=1,2,\dots$ , is polynomially bounded and lim⁡log_n⁡c_n(A_α)=3+α$\lim {\log }_n{c}_n(A_{\alpha })=3+\alpha$.

As an application we are able to construct a new example of a variety with an infinite basis of identities.