Strongly residual coordinates over A[x]

Abstract

For a commutative ring $A$, a polynomial $f \in A[x]^{[n]}$ is called a strongly residual coordinate if $f$ becomes a coordinate (over $A$) upon going modulo $x$, and $f$ becomes a coordinate (over $A[x, x^{-1}]$) upon inverting $x$. We study the question of when a strongly residual coordinate in $A[x]^{[n]}$ is a coordinate, a question closely related to the Dolgachev-Weisfeiler conjecture. It is known that all strongly residual coordinates are coordinates for $n = 2$ over an integral domain of characteristic zero. We show that a large class of strongly residual coordinates that are generated by elementaries over $A[x, x^{-1}]$ are in fact coordinates for arbitrary n, with a stronger result in the n = 3 case. As an application, we show that all Vénéreau-type polynomials are 1-stable coordinates.

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