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strong-real-jacobian

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Revision 280 · 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__strong-real-jacobian.md).”

Status and infobox

status: — → refuted
tier: — → V3
kind: — → conjecture
year: — → 1994
closed: — → 1994, S. Pinchuk
statement: — → Every polynomial map $F:\mathbb R^n\to\mathbb R^n$ with $\det JF(p)\neq0$ at every point $p$ is injective.
short: — → Nowhere-zero Jacobian on ℝⁿ implies injective — false from n = 2

Relations

+ implies real-keller: A constant nonzero determinant is in particular nowhere zero, so the strong real conjecture implies the real Jacobian question in every dimension.

Source +17 −0

+---

+title: The strong real Jacobian conjecture

+short: Nowhere-zero Jacobian on ℝⁿ implies injective — false from n = 2

+kind: conjecture

+status: refuted

+tier: V3

+year: 1994

+closed: 1994, S. Pinchuk

+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 [[implies::Statement:real-keller|the strong real conjecture implies the real Jacobian question in every dimension]].

+It is false. [[Statement:pinchuk-real|Pinchuk's maps]] 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).<ref>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).</ref>

+<references />

Revision 280 as rendered

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.

It is false. Pinchuk's maps 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]

↑ F. Braun, F. Fernandes, A counterexample to a folk real Jacobian conjecture, 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, arXiv:2605.12302 (May 2026).