Simple birational extensions of the polynomial algebra  |
| |
| Authors: | Shulim Kaliman Sté phane Vé né reau Mikhail Zaidenberg |
| |
| Affiliation: | Department of Mathematics, University of Miami, Coral Gables, Florida 33124 ; Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d'Hères cédex, France ; Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d'Hères cédex, France |
| |
| Abstract: | The Abhyankar-Sathaye Problem asks whether any biregular embedding  can be rectified, that is, whether there exists an automorphism  such that  is a linear embedding. Here we study this problem for the embeddings  whose image  is given in  by an equation  , where ![$finmathbb{C}[x,y]backslash{0}$](http://www.ams.org/tran/2004-356-02/S0002-9947-03-03398-1/gif-abstract0/img8.gif) and ![$ginmathbb{C}[x,y,z]$](http://www.ams.org/tran/2004-356-02/S0002-9947-03-03398-1/gif-abstract0/img9.gif) . Under certain additional assumptions we show that, indeed, the polynomial  is a variable of the polynomial ring ![$mathbb{C}^{[4]}=mathbb{C}[x,y,z,u]$](http://www.ams.org/tran/2004-356-02/S0002-9947-03-03398-1/gif-abstract0/img11.gif) (i.e., a coordinate of a polynomial automorphism of  ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings  . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial  as above, a criterion for when  . |
| |
| Keywords: | Affine space polynomial ring variable affine modification birational extension. |
|
| 点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息 |
|
点击此处可从《Transactions of the American Mathematical Society》下载全文 |
|
