A note on residual coordinates of polynomial rings

Abstract

A special case of the Dolgachev-Weisfeiler conjecture asserts that residual coordinates of the polynomial algebra $A =\mathbb{C} [x]^{[n]}$, $n\geq 3$, are coordinates. It is well known that such polynomials are stable coordinates; however, all the examples constructed thus far are actually 1-stable coordinates. In this paper, we show that all residual coordinates of $A $ are 1-stable coordinates.