English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the Zariski topology of the ind-groups of polynomial and freeassociative algebras $\Aut(K[x_1,...,x_n])$ (which is equivalent to theautomorphism group of the affine space $\Aut(K^n))$) and $\Aut(K< x_1,...,x_n>$ via $\Ind$-schemes, toric varieties, approximations and singularities.  We obtain some nice properties of $\Aut(\Aut(A))$, where $A$ is polynomial orfree associative algebra over a field $K$. We prove that all $\Ind$-schemeautomorphisms of $\Aut(K[x_1,...,x_n])$ are inner for $n\ge 3$, and all$\Ind$-scheme automorphisms of $\Aut(K< x_1,..., x_n>)$ are semi-inner. We alsoestablish that any effective action of torus $T^n$ on $\Aut(K< x_1,..., x_n>)$is linearizable provided $K$ is infinity. That is, it is conjugated to astandard one.  As an application, we prove that $\Aut(K[x_1,...,x_n])$ cannot be embeddedinto $\Aut(K< x_1,...,x_n>)$ induced by the natural abelianization. In otherwords, the {\it Automorphism Group Lifting Problem} has a negative solution.  We explore the close connection between the above results and the Jacobianconjecture, and Kontsevich-Belov conjecture, and formulate the Jacobianconjecture for fields of any characteristic.

On The Zariski Topology Of Automorphism Groups Of Affine Spaces And Algebras