The Abhyankar-Sathaye embedding conjecture

Theorem 1.1: an explicit $F\in\mathbb C[h,u,v,w]$, built from a perturbed cusp parametrisation, has $\mathbb C[h,u,v,w]/(F)\cong\mathbb C^{[3]}$ but a critical point on the fibre $F=-1$, so $F$ is not a coordinate; adjoining variables gives counterexamples for every $n\ge4$. The 5 October companion gives a degree-five $f$ in four variables with every fibre isomorphic to $\mathbb A^3$ and no fibre rectifiable, so the conjecture fails even for polynomials with all fibres affine. Not shown: the case $n=3$ (whether every embedded affine plane in $\mathbb A^3$ is rectifiable), which stays open.

Theorem 1.1: an explicit $F\in\mathbb C[h,u,v,w]$, built from a perturbed cusp parametrisation, has $\mathbb C[h,u,v,w]/(F)\cong\mathbb C^{[3]}$ but a critical point on the fibre $F=-1$, so $F$ is not a coordinate; adjoining variables gives counterexamples for every $n\ge4$. The 5 October companion gives a degree-five $f$ in four variables with every fibre isomorphic to $\mathbb A^3$ and no fibre rectifiable, so the conjecture fails even for polynomials with all fibres affine. Not shown: the case $n=3$ (whether every embedded affine plane in $\mathbb A^3$ is rectifiable), which stays open.