# Pinchuk's non-injective local diffeomorphisms of the plane

- kind: counterexample
- year: 1994
- authors: S. Pinchuk
- in one line: Real plane maps with positive, non-constant Jacobian that are not injective

**Statement.** There are polynomial maps $F=(P,Q):\mathbb R^2\to\mathbb R^2$ with $\det JF>0$ everywhere that are not injective. In the member of the class that Campbell analysed, $\deg P=10$, $\deg Q=25$, and $\mathbb R^2\setminus F(\mathbb R^2)$ is two points.

Pinchuk's maps are local diffeomorphisms of the plane that are not one-to-one. They [refute the strong real Jacobian conjecture, because their Jacobian is positive everywhere and they still fold the plane over itself](/wiki/Statement:strong-real-jacobian.md).[1] Their Jacobian is not constant, so they are not Keller maps and do not bear on JC₂ directly. What they show is that no proof of JC₂ can rest on real local invertibility and topology alone: [any proof of the plane conjecture has to use that the determinant is constant, or that the field is ℂ](/wiki/Statement:jc-plane.md).

Campbell drew one member of the class. The image misses exactly the two points $(0,0)$ and $(-1,-\tfrac{163}4)$, and its asymptotic variety is a connected curve, parametrized by polynomials of degrees 2 and 5.[2] That set is where the map fails to be proper, and Pinchuk's map is the standard worked example of a real [non-properness set](/wiki/Statement:jelonek-set.md).

## Relations

- refutes [strong-real-jacobian](/wiki/Statement:strong-real-jacobian.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. [1]
- related to [jc-plane](/wiki/Statement:jc-plane.md) — What they show is that no proof of JC₂ can rest on real local invertibility and topology alone: any proof of the plane conjecture has to use that the determinant is constant, or that the field is ℂ.

## References

1. S. Pinchuk, A counterexamle [sic] to the strong real Jacobian conjecture, Math. Z. 217 (1994) 1–4, [doi:10.1007/BF02571929](https://doi.org/10.1007/BF02571929).
2. L. A. Campbell, [Picturing Pinchuk's Plane Polynomial Pair](https://arxiv.org/abs/math/9812032), arXiv:math/9812032 (1998).

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