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On residual and stable coordinates
Journal of Pure and Applied Algebra ( IF 0.7 ) Pub Date : 2021-02-11 , DOI: 10.1016/j.jpaa.2021.106707
Amartya Kumar Dutta , Animesh Lahiri

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. 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,Z1,,Zn) to be a coordinate in k[X][Z1,,Zn,W]. Kahoui and Ouali had also shown that over a Noetherian d-dimensional ring R containing Q any residual coordinate in R[Z1,,Zn] is an r-stable coordinate, where r=(2d1)n. We will give a sharper bound for r when R is affine over an algebraically closed field of characteristic zero.



中文翻译:

关于残差和稳定坐标

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][ž1个žñ]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一种Xw ^+PXž1个žñ 成为一个坐标 ķ[X][ž1个žñw ^]。Kahoui和Ouali还表明,在一个Noetherian d维环R上 中的任何剩余坐标 [R[ž1个žñ]r稳定坐标,其中[R=2d-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。

更新日期:2021-02-15
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