Alpöge's polynomial map on ℂ³, whose Jacobian determinant is the constant −2, has no polynomial inverse.
Assessment
The evidence, examined directly, establishes the claim.
Alpöge's map sends (x, y, z) to a = (1+xy)³z + y²(1+xy)(4+3xy), b = y + 3x(1+xy)²z + 3xy²(4+3xy), c = 2x − 3x²y − x³z. Its Jacobian determinant expands to the constant −2. The map also takes the three distinct points (0, 0, −1/4), (1, −3/2, 13/2) and (−1, 3/2, 13/2) to the same image, (−1/4, 0, 0). A map that is not injective has no inverse of any kind, so it has no polynomial inverse. Both facts are short exact computations that anyone can repeat. More strongly, the map is generically three-to-one.
The Isabelle Archive of Formal Proofs entry proves the absence of a polynomial inverse directly. A Lean note machine-checks the triple collision. For polynomial self-maps of complex affine space, being non-injective and lacking a polynomial inverse amount to the same thing. The map is therefore a counterexample to the Jacobian conjecture in dimension three.
Full reasoning: the evidence and decisions behind this verdict
This pass recomputed both facts from the printed polynomials using symbolic algebra. The determinant of the Jacobian matrix expands exactly to −2. Each of the three rational points (0,0,−1/4), (1,−3/2,13/2) and (−1,3/2,13/2) evaluates to (−1/4, 0, 0). The step from a collision to no inverse is elementary: if G∘F = id, then F(p) = F(q) gives p = G(F(p)) = G(F(q)) = q. So a non-injective F has no left inverse, polynomial or not. That step needs no subclaim.
Independent corroboration comes from the verified claim that the map is generically three-to-one. That claim's own assessment used Gröbner bases and cites the Secret Blogging Seminar, Tao and Cook posts. It also comes from the Isabelle AFP entry (isa-afp.org/entries/Jacobian_Counterexample.html), which states that it proves the map has no polynomial inverse. That entry was not opened in this pass beyond its recorded passage.
The only residual doubt is whether the polynomials circulating in the sources are the ones that were announced. Every source the neighbouring claim examined prints the same ones. Nothing would change this verdict except a showing that the claim refers to a different map.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because the map sends three distinct points to a common image, it is not injective; any map with a left inverse, polynomial or not, is injective, so the map has no polynomial inverse.
- Does it follow?
- certain if the premises holdIf the premises are true, the conclusion must be. Whether they are is shown beside this.constitution →
- Do the premises hold?
- 1 not separately assessed
If the premise holds, the conclusion follows necessarily, because a map with any left inverse separates points. The argument rests entirely on the triple collision. That has not been assessed on its own page, but it is an exact rational computation: it was reproduced here, it is machine-checked in a Lean note, and it is entailed by the verified generic three-to-one property. That settled premise is what makes the argument decisive.
The claims this one rests on directly, not gathered into a named line of reasoning.
- supportsthis provides evidence for the parentsteward instructions →Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The theory defines polynomial invertibility and proves that the map has no polynomial inverse.
Refers to the explicit three-dimensional map announced by Levent Alpöge, whose Jacobian determinant is a nonzero constant.
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Created by extractor · Sep 16, 2026. Every judgment on this page is accompanied by a reasoning trace.