##### Department of Mathematics,

University of California San Diego

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### Mathematics Colloquium

## Ivan Shestakov

#### University of Sao Paulo, Brazil

## The Nagata automorphism is wild

##### Abstract:

It is well-known that the automorphisms of polynomial rings and free associative algebras in two variables are "tame", that is, they admit a decomposition into a product of linear automorphisms and the automorphisms of the type $(x,y)mapsto (x,y+f(x))$. However, in the case of three or more variables the similar question was open and known as ``The generation gap problem" or ``The tame generators problem". In 1972 Nagata constructed a certain automorphism of the polynomial ring in three variables and conjectered that it is non-tame or "wild". The purpose of the present work is to confirm the Nagata conjecture. Our main result states that the tame automorphisms of the polynomial ring in three variables over a field of characteristic $0$ are algorithmically recognizable. In particular, the Nagata automorphism is wild.

Host: Efim Zelmanov

### February 20, 2003

### 3:00 PM

### AP&M 6438

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