The growth of valuations on rational function fields in two variables |
| |
| Authors: | Edward Mosteig Moss Sweedler |
| |
| Affiliation: | Department of Mathematics, Loyola Marymount University, Los Angeles, California 90045 ; Department of Mathematics, Cornell University, Ithaca, New York 14853 |
| |
| Abstract: | ![]()
Given a valuation on the function field  , we examine the set of images of nonzero elements of the underlying polynomial ring ![$k[x,y]$](http://www.ams.org/proc/2004-132-12/S0002-9939-04-07456-8/gif-abstract0/img2.gif) under this valuation. For an arbitrary field  , a Noetherian power series is a map  that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on  . Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let  denote the images under the valuation  of all nonzero polynomials ![$f in k[x,y]$](http://www.ams.org/proc/2004-132-12/S0002-9939-04-07456-8/gif-abstract0/img8.gif) of at most degree  in the variable  . We construct a bound for the growth of  with respect to  for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight. |
| |
| Keywords: | Valuations, generalized power series, Gr" obner bases |
|
| 点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息 |
|
点击此处可从《Proceedings of the American Mathematical Society》下载全文 |
|
