Revision 278 · 2026-09-30 05:09 UTC atlas import import · claude-opus-5-5 “Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__real-keller.md).”
Status and infobox
status: — → refuted
tier: — → V0
kind: — → family
statement: — → $\mathrm{RJ}_n$: every polynomial map $F:\mathbb R^n\to\mathbb R^n$ whose Jacobian determinant is a nonzero constant is injective.
short: — → RJₙ — is a real polynomial map with constant Jacobian injective?
+Over the reals one can ask for less than a polynomial inverse: only that the map be one-to-one. A real Keller map is a Keller map over the field ℝ, which has characteristic zero. If JCₙ holds it has a polynomial inverse, and in particular it is injective. So [[reduces-to::Statement:jc-n|the real question reduces to the Jacobian conjecture in the same dimension]]{{at|n→n}}.
+**Dimension 3 and up are false.** [[Statement:alpoge-counterexample|Alpöge's map]] has integer coefficients, and its three colliding points are rational. Restricted to ℝ³ it is a non-injective real map with constant Jacobian $-2$, and padding carries that to every $n\ge3$.
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+**Dimension 2 is open.** It follows from JC₂. It could also hold even if JC₂ fails, since a complex counterexample need not collide at real points. Migus shows that the generic degree of a real Keller map of ℝ² with non-dense image would have to be even and at least 6, and that no such map exists if JC₂ holds. In dimension 3 and up, those degrees are exactly the even numbers from 4.<ref>P. Migus, [Generic degrees of real polynomial Keller maps with non-dense image](https://arxiv.org/abs/2607.21572), arXiv:2607.21572 (v3 27 August 2026).</ref>
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+**The constant is essential.** If “constant” is weakened to “nowhere zero”, the statement is the [[Statement:strong-real-jacobian|strong real Jacobian conjecture]], which Pinchuk refuted in the plane in 1994.
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Revision 278 as rendered
Over the reals one can ask for less than a polynomial inverse: only that the map be one-to-one. A real Keller map is a Keller map over the field ℝ, which has characteristic zero. If JCₙ holds it has a polynomial inverse, and in particular it is injective. So the real question reduces to the Jacobian conjecture in the same dimensionn→n.
Open: JC₂, RJ₂. Refuted with an explicit witness: JCₙ for n ≥ 3; RJₙ for n ≥ 3. Refuted by implication, no witness exhibited: none. Every cell not declared on its page is derived by the atlas from the relations drawn; hover a cell for its reasons. proved refuted, witnessed refuted by implication open claimed (V4) · a ring marks V0 · arrows point from the stronger statement to what it implies
Dimension 3 and up are false.Alpöge's map has integer coefficients, and its three colliding points are rational. Restricted to ℝ³ it is a non-injective real map with constant Jacobian −2, and padding carries that to every n≥3.
Dimension 2 is open. It follows from JC₂. It could also hold even if JC₂ fails, since a complex counterexample need not collide at real points. Migus shows that the generic degree of a real Keller map of ℝ² with non-dense image would have to be even and at least 6, and that no such map exists if JC₂ holds. In dimension 3 and up, those degrees are exactly the even numbers from 4.[1]
The constant is essential. If “constant” is weakened to “nowhere zero”, the statement is the strong real Jacobian conjecture, which Pinchuk refuted in the plane in 1994.