Adversarial checks and common objections

“The determinant is \(-2\), not \(1\)”

The standard hypothesis is any nonzero constant. In any case, the normalized map has determinant exactly \(1\), integer coefficients, and identity linear part.

“A nonzero Jacobian only gives local injectivity”

Correct—and the conjecture claimed that this local condition, for a polynomial self-map of affine space, forced global polynomial invertibility. The explicit collision disproves precisely that claim.

“The map is nonproper, so it is not a counterexample”

Properness is not a hypothesis of the Jacobian conjecture. Adding it would make the global conclusion much easier. The escape at infinity explains how the valid Keller map fails globally; see Discriminant and behavior at infinity.

“The three substitutions may miss a zero of the determinant”

The determinant was computed as a polynomial identity, not sampled at the three points. The hand proof works on a dense open set and then uses polynomial identity, while three CAS implementations independently expand it.

“Perhaps the three points are equal in disguise”

Their \(x\)-coordinates are \(0,1,-1\), so they are distinct over characteristic zero. The fiber Gröbner basis proves they are exactly the three reduced points over the target.

“Ax–Grothendieck or simple connectedness forbids this”

Ax–Grothendieck says an injective polynomial self-map is surjective; this map is not injective. Simple connectedness rules out nontrivial finite étale covers; this map is étale but nonproper and therefore not finite. Its image even omits an explicit curve.

“Known degree-three or prime-degree theorems forbid it”

The coordinate degrees are \((7,6,4)\). Cubic-homogeneous reductions raise dimension, and the recent prime field-extension result often mentioned in this context is explicitly two-dimensional. Neither applies to this map.

“A preprint already proved the conjecture”

The user-supplied arXiv:2209.01451 is an unrefereed claimed proof, not accepted literature. Its conclusion is contradicted by these exact identities. The history of this problem contains many withdrawn or corrected proof claims; see the literature audit.