theoremnovel

Noncoordinate polynomial with affine three-space zero fibre

R/(F)≅C[3]R/(F)\cong\CC^{[3]} but ∇⁡F\grad F vanishes at a point, so FF is no coordinate
the headline result
Theorem·thm:main·§1 Introduction and statement

Statement (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.

In plain English

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.

Why it's here

This is the paper's main result: a hypersurface that looks like a coordinate hyperplane algebraically but cannot be one. The isomorphism comes from the cusp and translation identities, the polynomial-ring description of the base algebra BB, and the lifting lemma applied to the unit ideal (h,y)(h,y). The critical point follows from the identity X2+Y3=s(1+F)X^2+Y^3=s(1+F), and a vanishing gradient contradicts the invertible Jacobian of any automorphism.
Classified novel:
This is the paper’s main new construction: an explicit polynomial in four variables whose zero fibre is affine three-space but which is not a coordinate. The lifting lemma establishes the polynomial-ring quotient, while the critical point rules out a coordinate; neither ingredient alone gives the combined example.

Uses

lemLifting lemma: polynomial algebra over B

Used by

corNoncoordinate polynomials in every dimension at least four

Inside the proof

claimCusp and translation identitiesclaimBase algebra is a polynomial ringclaimUnit ideal gives affine three-space fibreclaimAmbient identity gives critical pointclaimVanishing gradient excludes coordinate

Sources

PaperOpen
Section
§1 Introduction and statement
Page
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Label
thm:main
Line
paper.tex:103
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