On residual and stable coordinates

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][Z1,…,Zn] are one-stable coordinates. this we extend their result to the case arbitrary characteristic. fact, show holds when k[X] is replaced by any one-dimensional seminormal domain R which affine k. For our proof, S. Maubach [11] giving criterion for polynomial form a(X)W+P(X,Z1,…,Zn) be coordinate k[X][Z1,…,Zn,W]. had also shown Noetherian d-dimensional ring containing Q R[Z1,…,Zn] r-stable coordinate, where r=(2d−1)n. We will give sharper bound r zero.