# On Residual and Stable Coordinates

## TL;DR

This paper extends the understanding of residual coordinates in polynomial rings over algebraically closed fields, showing they are stable under broader conditions and providing sharper bounds for stability in certain rings.

## Contribution

It generalizes previous results to arbitrary characteristic and broader classes of rings, and refines bounds on the stability of residual coordinates.

## Key findings

- Residual coordinates are one-stable over broader classes of rings.
- Extended Maubach's criterion for polynomial coordinates.
- Provided sharper bounds for residual coordinate stability.

## Abstract

In a recent paper, 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 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 $Q$ 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.