On residual and stable coordinates

Article Type

Research Article

Publication Title

Journal of Pure and Applied Algebra


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=(2d−1)n. We will give a sharper bound for r when R is affine over an algebraically closed field of characteristic zero.



Publication Date



Open Access, Green

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