The stable coordinate conjecture in four variables

Theorem 1.1: with $Q=x_2^2-x_4^2+x_1x_3$ and $f=x_1-2Q(Q(x_2+x_4)+x_1x_4)$, a degree-five polynomial in $\mathbb C[x_1,\dots,x_4]$, some automorphism of $\mathbb C[x_1,\dots,x_4,w]$ sends $f$ to $x_1$ but none of $\mathbb C[x_1,\dots,x_4]$ does. Since the conjecture holds in at most three variables, this is the smallest possible dimension, and one added variable suffices. Corollary 1.2: every fibre of $f$ is $\mathbb A^3$ and none is rectifiable. Not shown: anything about dimensions above four, which the release's earlier five-variable example already covered for five; the result is one explicit polynomial, not a classification.

Theorem 1.1: with $Q=x_2^2-x_4^2+x_1x_3$ and $f=x_1-2Q(Q(x_2+x_4)+x_1x_4)$, a degree-five polynomial in $\mathbb C[x_1,\dots,x_4]$, some automorphism of $\mathbb C[x_1,\dots,x_4,w]$ sends $f$ to $x_1$ but none of $\mathbb C[x_1,\dots,x_4]$ does. Since the conjecture holds in at most three variables, this is the smallest possible dimension, and one added variable suffices. Corollary 1.2: every fibre of $f$ is $\mathbb A^3$ and none is rectifiable. Not shown: anything about dimensions above four, which the release's earlier five-variable example already covered for five; the result is one explicit polynomial, not a classification.