On residual and stable coordinates

Abstract

In a recent paper [10] , M.E. Kahoui and M. Ouali have proved that over an algebraically closed field k of characteristic zero, residual coordinates in k [ X ] [ Z 1 , … , Z n ] are one-stable coordinates. In this paper we extend their result to the case of an algebraically closed field k of arbitrary characteristic. In fact, we show that the result holds when k [ X ] is replaced by any one-dimensional seminormal domain R which is affine over an algebraically closed field k . For our proof, we extend a result of S. Maubach in [11] giving a criterion for a polynomial of the form a ( X ) W + P ( X , Z 1 , … , Z n ) to be a coordinate in k [ X ] [ Z 1 , … , Z n , W ] . Kahoui and Ouali had also shown that over a Noetherian d -dimensional ring R containing Q any residual coordinate in R [ Z 1 , … , Z n ] is an r -stable coordinate, where r = ( 2 d − 1 ) n . We will give a sharper bound for r when R is affine over an algebraically closed field of characteristic zero.