# A note on partial coordinate system in a polynomial ring

## TL;DR

This paper extends a result on partial coordinate systems in polynomial rings, showing that the property persists under more general conditions involving residual variables and arbitrary elements.

## Contribution

It generalizes previous theorems by incorporating the theory of residual variables to broader cases where the element is not necessarily a non-zerodivisor.

## Key findings

- Extension of Berson, Bikker, and van den Essen's result to arbitrary elements.
- Application of residual variables theory to polynomial rings.
- Broader conditions under which partial coordinate systems are preserved.

## Abstract

J. Berson, J. W. Bikker and A. van den Essen proved that for a non-zerodivisor $a$ in a commutative ring $R$ containing $Q$ if the polynomials $f_1,\dots,f_{n-1}$ in $R[X_1,\dots,X_n]$ form a partial coordinate system over the rings $R_a$ and $\dfrac{R}{aR}$ then $f_1,\dots,f_{n-1}$ form a partial coordinate system over the ring $R$. In this note we show that the theory of residual variables of Bhatwadekar-Dutta and its recent extension by Das-Dutta, extends their result to the case when $a$ is an arbitrary element of $A$.