David Azofeifa
A researcher facing two chalkboards covered with polynomial maps and curve sketches in a dark mathematics room

The Jacobian Conjecture for N = 2

An accessible August 2026 update on the two-variable Jacobian conjecture: what recent counterexamples changed, what they did not change, and why the complex plane case remains open.

· David Azofeifa · Mathematics

Tags: Jacobian Conjecture, Algebraic Geometry, Open Problems, Research Methods

Updated 19 August 2026. The classical two-variable Jacobian conjecture over the complex numbers is still open. Explicit counterexamples show that the conjecture fails in every dimension three and above, leaving the complex plane as the unresolved case.

The short version

The Jacobian conjecture asks a question about polynomial formulas. Suppose a formula takes points to points without ever locally folding, pinching, or tearing space. Must there be another polynomial formula that undoes it everywhere?

The answer is yes for one variable and no for three or more complex variables. The two-variable complex case remains open.

In July 2026, researchers exhibited a three-variable polynomial map that passes the conjecture’s local test but sends three distinct points to the same place. An independent formal proof has checked the decisive calculation. Adding unused coordinates turns that one example into counterexamples in every higher dimension.

None of that supplies a counterexample in two variables, and it does not prove that one cannot exist. JC(2), the complex-plane case, remains an open problem.

The puzzle without the jargon

Start with a familiar reversible polynomial rule:

F(x,y)=(x+y2, y).F(x,y)=(x+y^2,\ y).

If the output is (u,v)(u,v), recovering the input is easy:

F−1(u,v)=(u−v2, v).F^{-1}(u,v)=(u-v^2,\ v).

The rule may bend the plane, but it does not lose information. You can always work backward.

The Jacobian conjecture asks whether a particular local test guarantees that outcome for every polynomial rule. For a map F=(P,Q)F=(P,Q), the test is the determinant

det⁡JF=PxQy−PyQx.\det JF=P_xQ_y-P_yQ_x.

It measures the map’s infinitesimal change of area and orientation. When that determinant is a nonzero constant, the map looks reversible under a microscope at every point, with exactly the same local scale everywhere. Such a map is often called a Keller map.

The conjecture says:

If a polynomial map has a nonzero constant Jacobian determinant, then it has a polynomial inverse.

The necessary condition is easy to verify: if a polynomial inverse exists, ordinary differentiation forces the determinant to be a nonzero constant. The conjecture asks whether that condition is also sufficient to guarantee a polynomial inverse everywhere.

Why calculus does not settle it

The inverse-function theorem says that a nonzero Jacobian makes a map reversible near each individual point. That is a local promise, not a global one.

Imagine laying a transparent sheet over a map. If the sheet never creases or tears, then each tiny neighborhood can be flattened back out. But far away, the sheet might still disappear toward infinity and leave a gap in the overall picture. A local microscope cannot see that escape.

Mathematicians call the missing global condition properness. Roughly, a proper map does not allow points to run arbitrarily far away while their images remain bounded. A proper Keller map is reversible. The difficult possibility is a map that is locally flawless but loses information “at infinity.”

That is why the conjecture resisted an apparently obvious calculus argument for so long.

What changed in July 2026

An explicit map from complex three-dimensional space to itself was shown to have constant nonzero Jacobian determinant while taking three distinct points to the same output. That is enough to disprove the conjecture in dimension three: no inverse of any kind can send one output back to three different inputs.

The Archive of Formal Proofs verification checks, in Isabelle/HOL, both the constant determinant and the three-point collision. It also verifies the simple but important next step: appending identity coordinates produces counterexamples in every finite dimension n≥3n\ge3.

Soon after, Shuhong Gao’s preprint gave a self-contained geometric account and constructed further examples. Its main message is that the new phenomenon is not an isolated numerical accident: there are counterexamples in every dimension above two, and examples can have arbitrarily large generic fiber size. Here, “generic fiber size” just means how many inputs a typical output has.

This construction disproves the general conjecture, but it requires three coordinates. Removing one coordinate does not preserve the properties needed for a counterexample, so the construction leaves JC(2) unresolved.

Why the plane is still special

In two variables, curves and their behavior at infinity are much more tightly constrained than surfaces in three variables. Recent work has made that contrast sharper, but it has not removed it.

One useful example is T. Shaska’s July preprint on graded Keller maps. It proves that a two-variable Keller map with a compatible scaling symmetry is invertible. The result applies to that family; an arbitrary two-variable polynomial map need not have the required symmetry.

When a paper “solves a case” of the conjecture, check which maps it covers. A restricted result settles the whole question only if every possible map is known to belong to that case.

There is also a useful measure of how hard a hypothetical counterexample would have to be. A published degree-screening result rules out all but one degree pair below maximum degree 125 and raises the established lower bound from 100 to 108. In plain language: if a complex plane counterexample exists, it cannot be one of the comparatively simple formulas researchers have already been able to eliminate.

The bound narrows the possible counterexamples while leaving higher-degree maps to investigate.

A real plane counterexample that does not settle JC(2)

The terminology has become easier to mix up. In late July, Romy Mondello posted a preprint with a two-variable counterexample in characteristic two, the arithmetic setting built from the field with two elements. Its Jacobian determinant is one, yet three different points share an image.

That is an important result about the separable Jacobian conjecture in characteristic two. It is not a counterexample to the classical conjecture discussed here, which is about characteristic zero, especially complex numbers. Arithmetic changes dramatically when 1+1=01+1=0; a formula that works in characteristic two cannot simply be read as a complex counterexample.

To understand a result about the Jacobian conjecture, first check two things.

  • What dimension is the map in?

  • What number system is it using?

For the remaining open case, the answer must be two variables over characteristic zero.

What is actually known today

As of this update, the status can be stated cleanly.

Setting Status
One variable True, by elementary calculus and algebra.
Two variables over the complex numbers Open. No proof and no counterexample is established.
Three or more complex variables False. Explicit counterexamples exist.
Two variables in characteristic two A recent counterexample exists for a related separable version, not the classical problem.

The second row is unusually isolated. The conjecture is no longer a universal claim across all dimensions, but the plane may still obey the rule. We do not yet know whether it does.

How to read claims responsibly

This problem has a long history of convincing-looking false proofs. The recent breakthrough makes careful reading even more necessary, not less.

  • An explicit collision is decisive. If a constant-Jacobian map sends two distinct points to the same output, it cannot be invertible. The three-dimensional counterexample meets that standard, and its key facts have been formally checked.

  • A restricted theorem is still valuable. A result about symmetric, graded, low-degree, or otherwise special maps may be correct and useful without covering all Keller maps.

  • A preprint is evidence, not the end of the process. It deserves to be read, checked, discussed, and eventually either strengthened or corrected. The sources below identify preprints as such.

  • “Two-dimensional” is incomplete on its own. Characteristic two and characteristic zero are different problems.

These checks help distinguish what a result establishes from the questions it leaves open.

Why this remaining question matters

The Jacobian conjecture is a compact question about polynomial formulas, but it sits where algebra, geometry, and the local-versus-global divide meet. It asks when perfect behavior in every tiny neighborhood is enough to control an entire infinite space.

The 2026 counterexamples changed the story from “this should always be true” to a more interesting pair of questions:

  1. Why can information escape to infinity in three dimensions and above?

  2. What, if anything, prevents the same escape in the complex plane?

An answer to the second question could go either way. It might prove that every two-variable Keller map really is reversible. Or it might reveal the first complex planar counterexample. At present, both outcomes remain possible.

Sources and further reading