# The real Jacobian question in dimension n

- kind: family
- status: refuted (evidence tier V0)
- in one line: RJₙ — is a real polynomial map with constant Jacobian injective?

**Statement.** $\mathrm{RJ}_n$: every polynomial map $F:\mathbb R^n\to\mathbb R^n$ whose Jacobian determinant is a nonzero constant is 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 [the real question reduces to the Jacobian conjecture in the same dimension](/wiki/Statement:jc-n.md).

| | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 |
|---|---|---|---|---|---|---|
| JCₙ | proved | open | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |
| RJₙ | proved | open | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |

**Dimension 3 and up are false.** [Alpöge's map](/wiki/Statement:alpoge-counterexample.md) 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 $n\ge3$.

**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](/wiki/Statement:strong-real-jacobian.md), which Pinchuk refuted in the plane in 1994.

## Relations

- reduces to [jc-n](/wiki/Statement:jc-n.md) — So the real question reduces to the Jacobian conjecture in the same dimension.

## References

1. 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).

## Relations to this page

- refuted by [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md) — So the restriction to ℝ³ is a real polynomial map with constant Jacobian that is not injective, and the map also refutes the real Jacobian question in dimension three.
- implied by [strong-real-jacobian](/wiki/Statement:strong-real-jacobian.md) — A constant nonzero determinant is in particular nowhere zero, so the strong real conjecture implies the real Jacobian question in every dimension.

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