# Invertible Polynomial Transformations

## Document Type

Presentation Abstract

## Presentation Date

4-22-1999

## Abstract

The *general affine group* *GA _{n}*(C) generalizes the more familiar general linear group

*GL*(C). In particular,

_{n}*GA*(C) is the set of one-to-one functions

_{n}*F*= (

*F*

_{1}, …,

*F*

_{n}) : C

^{n}→ C

^{n}such that each

*F*is a polynomial in

_{i}*n*variables over C. Remarkably, such

*F*are also onto, and

*F*

^{-1}is an element of

*GA*(C). So

_{n}*GA*(C) is indeed a group, consisting of the invertible polynomial transformations of C

_{n}^{n}, with

*GL*(C) as a subgroup. The aim of this talk is to give an overview of what is known about this important and much-studied group.

_{n}The *Structure Theorem* for *GA _{2}*(C) gives a fairly complete understanding in this case. For

*n*>= 3, relatively little is known about the structure of

*GA*(C), except that it is amazingly complicated. For example, the

_{n}*tame*subgroup

*T*contained in

_{n}*GA*(C) is easy to define, and the Structure Theorem implies

_{n}*T*

_{2}=

*GA*(C), but it remains an open question whether

_{2}*T*=

_{n}*GA*(C) for any

_{n}*n*>= 3.

Naturally, one wishes to study certain kinds of group actions in which an "algebraic" group *G* acts "algebraically" on C^{n}, since these give rise to embeddings of *G* as a subgroup of *GA _{n}*(C). Classically, the case in which

*G*is a

*reductive*group (like

*G*=

*SL*(C)) has been studied since the Nineteenth Century, and many positive results are known. The case in which

_{n}*G*is a

*unipotent*group (like

*G*= C

^{+}, the additive group of C) is not as well understood, though the importance of this case is widely recognized. Much of my own work has focused on actions of C

^{+}on C

^{n}.

## Recommended Citation

Freudenburg, Dr. Gene, "Invertible Polynomial Transformations" (1999). *Colloquia of the Department of Mathematical Sciences*. 43.

https://scholarworks.umt.edu/mathcolloquia/43

## Additional Details

Thursday, 22 April 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)