The Zariski cancellation problem for complex affine space in dimension four

Theorem 1.1: with $P=\mathbb C[p,s,u,F,J]$, $x=s^2+u^3+p^2F$ and $H=x^2F-(1+2sx)J-p^2J^2-pu$, the algebra $A=P/(H)$ is an integral complex domain of dimension four with $A[w]\cong\mathbb C^{[5]}$ but $A\not\cong\mathbb C^{[4]}$. The stabilization uses that $H$ is a residual coordinate (El Kahoui-Ouali, Dutta-Lahiri); non-polynomiality is proved by degenerating to a graded algebra, lifting a putative additive action through a torus bundle over a quadric, and excluding it by a Mason-Stothers abc argument. It answers only $n=4$: dimension three is not treated, and the paper notes that adjoining variables to a counterexample does not by itself give counterexamples in higher dimensions.

Theorem 1.1: with $P=\mathbb C[p,s,u,F,J]$, $x=s^2+u^3+p^2F$ and $H=x^2F-(1+2sx)J-p^2J^2-pu$, the algebra $A=P/(H)$ is an integral complex domain of dimension four with $A[w]\cong\mathbb C^{[5]}$ but $A\not\cong\mathbb C^{[4]}$. The stabilization uses that $H$ is a residual coordinate (El Kahoui-Ouali, Dutta-Lahiri); non-polynomiality is proved by degenerating to a graded algebra, lifting a putative additive action through a torus bundle over a quadric, and excluding it by a Mason-Stothers abc argument. It answers only $n=4$: dimension three is not treated, and the paper notes that adjoining variables to a counterexample does not by itself give counterexamples in higher dimensions.