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 $k[X][Z_1,\dots ,Z_n]$ 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 $a(X)W+P(X,Z_1,\dots ,Z_n)$ to be a coordinate in $k[X][Z_1,\dots ,Z_n,W]$. Kahoui and Ouali had also shown that over a Noetherian d-dimensional ring R containing $\mathbb{Q}$ any residual coordinate in $R[Z_1,\dots ,Z_n]$ 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.

在最近的一篇论文[10]中，ME Kahoui和M. Ouali证明了在特征为零的代数封闭场k上，剩余坐标为$ķ[X][{ž}_{\mathrm{1个}}，\dots ，{ž}_{ñ}]$是单稳态坐标。在本文中，我们将其结果扩展到任意特征的代数闭合场k的情况。实际上，我们表明结果保持$ķ[X]$用代数封闭场k上仿射的任何一维半正规域R代替。为证明起见，我们扩展了[11]中S. Maubach的结果，给出了形式为多项式的准则$\mathrm{一种}（X）\mathrm{w\; ^}+P（X，{ž}_{\mathrm{1个}}，\dots ，{ž}_{ñ}）$ 成为一个坐标 $ķ[X][{ž}_{\mathrm{1个}}，\dots ，{ž}_{ñ}，\mathrm{w\; ^}]$。Kahoui和Ouali还表明，在一个Noetherian d维环R上，$\mathbb{问}$ 中的任何剩余坐标 $\mathrm{[R}[{ž}_{\mathrm{1个}}，\dots ，{ž}_{ñ}]$是r稳定坐标，其中$\mathrm{[R}=（2^d-\mathrm{1个}）ñ$。当R在特征零的代数封闭场上仿射时，我们将为r给出一个更清晰的边界。