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 16303131&_mathId=si1.gif&_user=111111111&_pii=S0021869316303131&_rdoc=1&_issn=00218693&md5=a9723460f0068d8462ee8bf21db4627e" title="Click to view the MathML source">α∈(0,1)$\alpha \in (0,1)$ a commutative nonassociative algebra 16303131&_mathId=si104.gif&_user=111111111&_pii=S0021869316303131&_rdoc=1&_issn=00218693&md5=bed1c8a4e6abb16aba956b1cc8e5e179" title="Click to view the MathML source">A_α$A_{\alpha }$ whose codimension sequence 16303131&_mathId=si3.gif&_user=111111111&_pii=S0021869316303131&_rdoc=1&_issn=00218693&md5=22198f13359863c0e17a47eae2fb8c5f" title="Click to view the MathML source">c_n(A_α)$c_n(A_{\alpha })$, 16303131&_mathId=si4.gif&_user=111111111&_pii=S0021869316303131&_rdoc=1&_issn=00218693&md5=2b1615f9106fd1ef447d3f3c320c6369" title="Click to view the MathML source">n=1,2,&hellip;$n=1,2,\&<font\; color="red">hellip</font>;$ , is polynomially bounded and 16303131&_mathId=si5.gif&_user=111111111&_pii=S0021869316303131&_rdoc=1&_issn=00218693&md5=016092d556caa3d59a2fef9f7e2f7d61" title="Click to view the MathML source">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.