A note on k[z]-automorphisms in two variables
| dc.creator | Edo, Eric | |
| dc.creator | Essen, Arno van den | |
| dc.creator | Maubach, Stefan | |
| dc.date | 2008-09-04 | |
| dc.date.accessioned | 2026-07-07T10:00:40Z | |
| dc.date.available | 2026-07-07T10:00:40Z | |
| dc.description | We prove that for a polynomial $f\in k[x,y,z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x,y,z]/(f)\cong k^{[2]}$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a\in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f\in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x,y,z]$, and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method - essentially linking Nagata with a non-tame $R$-automorphism of $R[x]$, where $R=k[z]/(z^2)$. | |
| dc.description | 8 pages | |
| dc.identifier | https://arxiv.org/abs/0809.0767 | |
| dc.identifier | http://arxiv.org/abs/0809.0767 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/168382 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14R10, 13B25, 14J50 | |
| dc.title | A note on k[z]-automorphisms in two variables | |
| dc.type | text |