Stable tameness of automorphisms of $${F\langle x,y,z\rangle}$$ fixing z

What is it about?

Whether every automorphism of a free associative algebra (polynomial algebra) over a field $F$ is stably tame is a long-standing open question. An $F$-automorphism of a free associative algebra $F\langle x_1,\ldots,x_n\rangle$ (a polynomial algebra $F[x_1,\ldots,x_n]$) is tame if it is the product of elementary automorphisms. An $F$-automorphism $(f_1,\ldots,f_n)$ of $F\langle x_1,\ldots,x_n\rangle$ ($F[x_1,\ldots,x_n]$) is stably tame if there exists a nonnegative integer $m$ such that the automorphism $(f_1,\ldots, f_n,x_{n+1},\ldots, x_{n+m})$ of $F\langle x_1,\ldots,x_{n+m}\rangle$ ($F[x_1,\ldots,x_{n+m}]$) is tame. A polynomial $f\in F\langle x_1,\ldots,x_n\rangle$ is a stably tame coordinate if the automorphism $(f,f_2,\ldots,f_n)$ is stably tame, for some $f_2,\ldots,f_n\in F\langle x_1,\ldots,x_n\rangle$. If there exists an automorphism $(f,f_2,\ldots,f_{n-1},x_n)$ then $f$ is an $x_n$-coordinate. In the paper under review the authors prove that every fixing $z$ automorphism of the free associative algebra $F\langle x,y,z\rangle$ over an arbitrary field $F$ is stably tame and becomes tame after adding one variable. As a consequence, they obtain that every $z$-coordinate of $F\langle x,y,z\rangle$ is stably tame. Reviewed by Érica Z. Fornaroli

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