# The strong real Jacobian conjecture

- kind: conjecture
- status: refuted (evidence tier V3)
- year: 1994
- in one line: Nowhere-zero Jacobian on ℝⁿ implies injective — false from n = 2

**Statement.** Every polynomial map $F:\mathbb R^n\to\mathbb R^n$ with $\det JF(p)\neq0$ at every point $p$ is injective.

Over ℂ a polynomial that never vanishes is constant, so “nowhere zero” and “constant” are the same hypothesis. Over ℝ they are not, and the weaker hypothesis gives a stronger conjecture. A constant nonzero determinant is in particular nowhere zero, so [the strong real conjecture implies the real Jacobian question in every dimension](/wiki/Statement:real-keller.md).

It is false. [Pinchuk's maps](/wiki/Statement:pinchuk-real.md) of ℝ² have everywhere positive Jacobian and are not injective, and padding with the identity carries the failure to every $n\ge2$. The conjecture holds for $n=1$, where a nowhere-zero derivative makes a polynomial strictly monotone.

Later work sharpened the example. Braun and Fernandes (2025) show that even a Jacobian bounded away from zero does not force injectivity. Braun, Gwoździewicz, Fernandes and Oréfice-Okamoto show that injectivity does follow when $\deg p\le6$, and give a non-injective example with $\deg p=7$ (unrefereed).[1]

## Relations

- implies [real-keller](/wiki/Statement:real-keller.md) — A constant nonzero determinant is in particular nowhere zero, so the strong real conjecture implies the real Jacobian question in every dimension.

## References

1. F. Braun, F. Fernandes, [A counterexample to a folk real Jacobian conjecture](https://doi.org/10.1007/s11856-025-2726-1), Israel J. Math. 267 (2025) 867–870. F. Braun, J. Gwoździewicz, F. Fernandes, B. Oréfice-Okamoto, [A sharp degree bound in the real Jacobian conjecture](https://arxiv.org/abs/2605.12302), arXiv:2605.12302 (May 2026).

## Relations to this page

- refuted by [pinchuk-real](/wiki/Statement:pinchuk-real.md) — They refute the strong real Jacobian conjecture, because their Jacobian is positive everywhere and they still fold the plane over itself. Their Jacobian is not constant, so they are not Keller maps and do not bear on JC₂ directly.

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