theoremnovel
Noncoordinate polynomial with affine three-space zero fibre
but vanishes at a point, so is no coordinate
the headline result
Theorem·
thm:main·§1 Introduction and statementStatement (paper)
The polynomial defined in -- satisfies
Consequently is not a coordinate of .
In plain English
The explicit polynomial in four variables satisfies , so its zero fibre is affine three-space, yet . Therefore is not a coordinate of .
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 , and the lifting lemma applied to the unit ideal . The critical point follows from the identity , 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.