GPE
Family 049Algebraic and complex geometryLeanSep 24, 2026

An explicit noncoordinate polynomial with affine three-space zero fibre

What this paper proves

An explicit F∈C[h,u,v,w]F\in\mathbb{C}[h,u,v,w] satisfies C[h,u,v,w]/(F)≅C[3]\mathbb{C}[h,u,v,w]/(F)\cong\mathbb{C}^{[3]} and ∇F(2,0,−1/2,1/2)=0\nabla F(2,0,-1/2,1/2)=0, so it is not a coordinate. An elementary lifting lemma converts cusp and translation identities into polynomial coordinates on the zero fibre. This disproves the Abhyankar–Sathaye conjecture for hypersurfaces in complex affine four-space.

Main theorem

theoremnovelNoncoordinate polynomial with affine three-space zero fibrethm:main
The explicit polynomial FF in four variables (h,u,v,w)(h,u,v,w) satisfies R/(F)≅C[3]R/(F)\cong\CC^{[3]}, so its zero fibre is affine three-space, yet ∇⁡F(2,0,−1/2,1/2)=0\grad F(2,0,-1/2,1/2)=0. Therefore FF is not a coordinate of RR.
Theorem 1.1 · as stated in the paper
The polynomial FF defined in -- satisfies R/(F)≅C[3],∇⁡F(2,0,−1/2,1/2)=0. R/(F)\cong\CC^{[3]}, \qquad \grad F(2,0,-1/2,1/2)=0. Consequently FF is not a coordinate of RR.

Where the new idea lives · 1 novel step (one is the main theorem)

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