The Jacobian Conjecture for N = 2

The Jacobian conjecture asks whether a polynomial rule that can be reversed near every input must have a polynomial reverse rule that works everywhere. Its difficulty lies in the difference between a tiny neighborhood and an entire infinite space.

Updated 9 October 2026. The conjecture is true for one complex variable, false for three or more, and still open for two. The investigation described here has found useful restrictions and failed approaches, but no proof or counterexample in two variables. Its newest general arguments still need independent review.

We will start with a rule you can undo, use the pictures to understand the three-variable counterexample, and then explore why two variables remain unresolved. The later sections explain what the research attempts have taught us. The full three-variable formula is in a short optional section at the end.

A rule you can run backward

A polynomial is a formula built from constants and variables using addition, subtraction, multiplication, and nonnegative whole-number powers. For example, $x+y^2$ is a polynomial; $1/x$ is not, because it divides by a variable.

Consider a rule that takes two input coordinates and returns two output coordinates:

Mathematicians call such a rule a map. This one adds the square of the second coordinate to the first, while leaving the second alone. The input $(2,3)$ becomes $(11,3)$.

To run it backward, read the second output and subtract its square from the first. If we name the outputs $u$ and $v$, the inverse rule is

The superscript $-1$ means “the inverse rule” here. It does not mean division by $F$.

Figure 1. A bend you can undo. The highlighted point moves from $(0,1)$ to $(1,1)$. Every row keeps its height, so subtracting the same amount restores the original grid. Both panels use the same scale. Enlarge the grid diagram.

The conjecture concerns complex numbers, which include numbers involving $i$, where $i^2=-1$. One complex coordinate has a real part and an imaginary part. Two complex coordinates therefore contain four real coordinates, more than this flat picture can show. The drawing shows the real-coordinate slice; the inverse formula works for complex inputs too.

Throughout the article, “two variables” means two complex coordinates unless a different setting is stated. The notation $\mathbb C^2$ in the figures is shorthand for this space of pairs of complex numbers.

What the Jacobian checks

Imagine zooming in on a tiny patch of the grid. Calculus describes the map there by a linear approximation: a combination of stretching, turning, and shearing. The Jacobian determinant tests whether that approximation flattens any direction. A nonzero value guarantees that the map can be reversed in a sufficiently small neighborhood.

Write a two-output map as $F=(P,Q)$. Its Jacobian determinant is

The subscripts mean partial derivatives: for example, $P_x$ measures how $P$ changes when $x$ changes while $y$ stays fixed. You do not need to calculate derivatives to follow the rest of the article. What matters is the test: the determinant must be the same nonzero number at every point.

Our grid example passes. Its determinant is $1$ everywhere. A map with polynomial coordinates that passes this test is called a Keller map, a term that also appears in some of the figures.

The remaining two-variable conjecture says:

In two complex variables, every polynomial map with a nonzero constant Jacobian determinant has a polynomial inverse defined everywhere.

Having a constant determinant does not mean preserving every distance or shape. Figure 1 bends vertical lines. For real coordinates, the determinant measures local signed area scaling; the complex version supplies the same local reversibility test.

In one variable, the conjecture is easy. A polynomial with constant nonzero derivative must be $ax+b$, with $a\ne0$, and its inverse is $(u-b)/a$. With two variables, the coordinates can interact in much more complicated ways.

Why calculus does not settle it

The inverse-function theorem gives a reverse rule near each individual input. It does not promise that distant inputs have different outputs, or that every output is reached.

Infinity creates another difficulty. Inputs might travel arbitrarily far away while their outputs settle near an ordinary, finite point. Calculus at any one input cannot see the whole journey.

The condition that prevents this escape is called properness. In our setting, a proper map cannot have inputs escaping every bounded region while their outputs remain bounded. A polynomial map with both properness and the nonzero constant Jacobian condition has a polynomial inverse. Proving properness for every two-variable Keller map would therefore settle JC(2).

Here is a simple illustration of escape. Temporarily allow division and use

This map also has Jacobian determinant $1$. Feed it the inputs $(t,1/t^2)$ and let the positive number $t$ grow. The outputs are $(1/t,-1)$: the inputs run away, but the outputs approach $(0,-1)$.

Figure 2. A local test cannot see infinity. The first input keeps growing while its output gets closer to zero. The second output stays at $-1$, so the full output stays bounded. The panels use different vertical scales. Enlarge the escape graph.

A formula that allows division by polynomials is called rational. Its domain is the set of inputs where it is defined. Here that excludes the line $x=0$, so this is not a counterexample to the polynomial conjecture. It shows the kind of behavior a proof must control, while also showing why the domain matters.

What changed in July 2026

Researchers found a three-variable polynomial map with determinant $-2$ everywhere that sends three distinct inputs to the same output. Call it $F_3$. A shared output like this is a collision: it makes a reverse rule impossible because there is no unique input to recover.

The Archive of Formal Proofs independently verified both the determinant identity and the collision using Isabelle/HOL, software that checks proofs written as precise logical steps. Appending extra coordinates that simply pass through unchanged gives counterexamples in every higher dimension too.

Three inputs, one output

The collision uses ordinary fractions, so its inputs are easy to state even though the full formula is long.

Figure 3. The decisive collision. The points $(0,0,-1/4)$, $(1,-3/2,13/2)$, and $(-1,3/2,13/2)$ all map to $(-1/4,0,0)$. The arrows mean “has this output”; they are not paths through space. The determinant is nonzero everywhere, so the rule passes the local test despite this collision. Enlarge the collision diagram.

These are two separate facts. Substituting three points checks the collision. Establishing the constant determinant requires an identity valid for every input. The formal verification checks both.

Why tangents help create a collision

Shuhong Gao’s geometric account explains the construction through tangent lines. A tangent is the line that follows a curve’s direction at a chosen point. Different tangents to the same curve can pass through one shared point.

An intermediate rule chooses a point on a cubic curve, then chooses a position along its tangent. If several tangents reach the same destination, several choices produce the same output.

Figure 4. Three routes to the same point. The curve is labeled by a parameter $w$. The tangents at $w=-1,1,2$ meet at $(-1,-1)$. The symbol $\gamma$ in the figure describes position along a tangent. This intermediate map has determinant $2\gamma$, which becomes zero when $\gamma=0$. Enlarge the tangent graph.

That last sentence explains why the drawing alone is not a two-variable counterexample. The construction needs an additional coordinate and exact algebraic cancellations to produce polynomial formulas with a constant nonzero determinant. The extra coordinate has a job to do.

Where the extra inputs go

The same verified map also lets us watch escape happen. Let $t$ increase from $1$. There are three inputs, named $p_0(t)$, $p_+(t)$, and $p_-(t)$, that all have output

The first coordinate of $p_+(t)$ is $t$, and that of $p_-(t)$ is $-t$. Those two inputs move farther away in opposite directions. The third input approaches the origin, where all coordinates are zero.

Figure 5. Two inputs disappear toward infinity. As $t$ grows, the shared output approaches $(0,0,0)$. At that limiting output, the only input is $(0,0,0)$. The two other inputs have escaped every bounded region; they have not merged at an ordinary point. The complete input formulas appear in the optional section below. Enlarge the escape diagram.

This is how local reversibility can survive a change in the number of inputs. There is no finite meeting point where the derivative must fail. The missing inputs leave through infinity.

Why two variables remain open

A natural response is to take a slice with two free coordinates through the three-variable example. Fix one input coordinate, let the other two vary, and retain two of the outputs. This creates a two-variable rule, but it may lose one of the properties we need.

Here are two exact attempts, both keeping the first two outputs of $F_3$:

• Fix the input $x=0$. The remaining rule is $(y,z)\mapsto(z+4y^2,y)$. It passes the Jacobian test, but it is reversible: the second output reveals $y$, and then the first reveals $z$.
• Fix the input $z=13/2$. Two of the colliding inputs remain. However, the new determinant is zero at $(x,y)=(0,0)$, so the rule fails the Jacobian test.

Figure 6. Each slice loses something essential. The left slice has constant nonzero determinant but no collision. The right slice keeps a collision but has a zero determinant at the origin. These calculations exclude these two attempts; they do not exclude every possible two-variable construction. Enlarge the slice comparison.

A counterexample in two variables must satisfy all the requirements together: polynomial formulas defined on the whole space, a constant nonzero Jacobian, and failure of a global inverse. A proof must establish an inverse for every map satisfying the hypothesis.

Figure 7. The missing step. The arrow to a local inverse is known. The arrow to an inverse that works everywhere is still open in two variables. Proving properness for every such map would bridge that gap by excluding escape with bounded outputs. The three-variable counterexample does not tell us whether the same escape is possible with only two variables. Enlarge the open-question diagram.

Two variables and characteristic two mean different things

Some headlines concern a different “two.” Characteristic two is an arithmetic setting in which $1+1=0$. It changes how polynomial formulas and derivatives behave.

Romy Mondello’s preprint gives a two-variable counterexample to a related separable Jacobian conjecture in characteristic two. The classical question in this article uses complex numbers, where $1+1=2$. A construction in the first setting cannot simply be transferred to the second.

The distinction gives this status table for the 9 October update:

Setting	What is known
One complex variable	Every map passing the test has a polynomial inverse.
Two complex variables, JC(2)	Open: no proof or counterexample is established here.
Three or more complex variables	Explicit counterexamples have been verified.
Two variables in characteristic two	A counterexample exists for a related version using different arithmetic.

Two different meanings of degree

Before looking at the research attempts, we need to separate two measures that are both called degree.

The polynomial degree measures the powers in a formula. In several variables, add the exponents in each term and take the largest total: $x^2y^3$ has degree $5$.

The other measure counts how many inputs a typical output has. “Typical” means away from exceptional output sets where the count can change, as it did in Figure 5. Researchers call this the generic fiber size or generic degree. A fiber is simply the collection of inputs with a specified output.

Figure 8. Bigger powers do not necessarily mean more inputs. In the family $F_k(x,y)=(x+y^k,y)$, the polynomial degree is $k$. Every output $(u,v)$ still has exactly one input, $(u-v^k,v)$. The bars compare $k=2,4,8$; the label $N$ denotes the input count. Enlarge the degree chart.

This distinction matters when judging progress. Making a formula look complicated does not necessarily create collisions. Even a reversible change of coordinates can increase polynomial degrees while leaving the input count unchanged. A proposed bound must specify which degree it controls.

There are substantial results about formula degree. A 2022 paper by Guccione, Guccione, Horruitiner, and Valqui shows that a hypothetical planar counterexample must have maximum polynomial degree at least $108$. Below $125$, its only remaining degree pair is $(72,108)$, up to swapping the two outputs.

An August manuscript and supplement claiming a bound of 125 was also inspected during this investigation. Its supporting calculations were not rerun and its proof was not independently reconstructed, so that stronger claim is not treated as verified here. Neither bound, even if established, settles what happens at all higher degrees.

Reconsidering dimensional shadows and projections

A projection throws information away. Two points at different heights can cast the same shadow.

Figure 9. A shared shadow is easy to arrange. Both points project to $(1,1)$ when the third coordinate is discarded. This ordinary three-dimensional drawing illustrates information loss; it does not itself construct a two-variable Keller map. Enlarge the projection diagram.

The challenge is to preserve a collision and the Jacobian condition in the same two-variable polynomial rule. The determinant of a three-variable map does not automatically determine the determinant of its shadow.

The research notes rule out one broad family of attempts with the verified $F_3$: substitute any polynomial $z=h(x,y)$, then form two outputs by adding fixed multiples of the original three outputs. In that family, the new determinant can never be a nonzero constant. This covers those particular surfaces and projections, not every way to reduce dimension.

Another approach tries to place a complete copy of the two-variable space inside both the input and output spaces, with $F_3$ mapping one copy directly into the other. Such a restriction can have at most three inputs per output. The established planar degree-three theorem then rules out a counterexample of this kind. Here “degree three” refers to the input count, not the powers in the formulas.

These failures explain why the extra coordinate cannot simply be erased. Other constructions require their own arguments, including proof that the proposed surface really is the whole two-variable space rather than a space with points or curves missing.

Reconsidering the Riemann sphere and the Möbius strip

The Riemann sphere gives infinity a place in the picture. Start with one complex coordinate, which we can draw as an ordinary plane, and add a single point at infinity. The resulting space can be viewed as a sphere.

Figure 10. Bringing infinity into view. In this drawing, zero lies at the south pole and $-1$ and $1$ lie on the equator. Both ends of the real axis approach the added point at the north pole. The highlighted meridian represents only the real axis; the whole sphere represents one complex coordinate. Enlarge the sphere diagram.

This process is called compactification. It lets mathematicians study an escaping path by examining its limiting behavior at an added boundary. For two complex coordinates, a standard compactification adds an entire curve at infinity. The geometry is richer than a single sphere.

The sphere suggests useful rational maps. A simple example of the construction examined in the notes is

Its determinant is $1$, and the distinct inputs $(x,y)$ and $(-x,-y)$ have the same output. It looks promising until we notice the division by $x$.

Trying to remove that division exposes the problem. Substitute $y=2xt$. The resulting rule is $(x,t)\mapsto(x^2,t)$, whose determinant is $2x$ and vanishes at $x=0$. The substitution was not reversible there. Removing the denominator has not produced a polynomial counterexample.

More elaborate attempts build larger surfaces containing the whole two-variable space and an added boundary. Some candidate surfaces pass useful geometric checks. The remaining task is to find formulas on them that fit together globally and satisfy the Jacobian identity at every ordinary input. A suitable surface alone does not supply those formulas.

The Möbius strip offers a different suggestion: traveling around a loop might exchange two apparent choices. Its literal reversal of orientation is not the geometry of a smooth complex curve, which has a consistent orientation. The useful idea is to follow several solutions around a loop and see how they exchange. We will make that precise with square roots in Figure 13.

What the proof attempts have taught us

The investigation has made progress in restricted settings. That means an argument covers a specified class of maps, with stated extra assumptions. It does not mean every possible counterexample belongs to that class.

One line of work expands the formulas and compares coefficients, the numbers or expressions multiplying each power. In a particular family, the necessary equations force the determinant to be zero rather than one, so that family cannot contain a counterexample. Extending the coefficient calculation covered some additional terms, but not every possible polynomial form.

Another line of work adds the missing boundary algebraically and asks which properties of the enlarged space would force an inverse. This produced stronger conditional criteria. Their unresolved part is proving that every Keller map meets the extra conditions.

The boundary constructions also produced useful failures. Some surfaces have the right open plane inside them, yet their simplest coordinate formulas cannot pass the Jacobian test. Other formulas agree with the required behavior to any chosen finite precision near the boundary, while still failing at ordinary points. In those examples, higher precision also requires higher polynomial degree.

These arguments help identify where a proposed proof needs more work. The detailed derivations and exact experiments remain in the companion research notes. The newest general deductions have not received independent review. The following two examples show both the value and the limits of this kind of progress.

Lines, curves, and collision limits

Suppose two moving inputs always have the same output and both inputs approach finite endpoints. For a polynomial Keller map, those endpoints must still be a valid pair of distinct inputs with the same output.

There are two reasons. Polynomials are continuous, so equal outputs remain equal in the limit. And local reversibility prevents the two inputs from merging at a single point: sufficiently near that point, the reverse rule would make them identical already. The research notes call this requirement closedness of the collision relation.

The latest argument combines this limit requirement with two differential expressions, $P\,dQ$ and $Q\,dP$. Here $dQ$ records the change in $Q$ in each input direction, and $P\,dQ$ multiplies that change by $P$. For polynomial outputs, the coefficients in both expressions are polynomial. Rational outputs can sometimes pass this check too, because denominators cancel.

The argument examines maps with determinant $1$ for which both expressions are polynomial and the full collision relation is closed, including all finite limits. It then considers the case where the product $PQ$ factors entirely into linear factors. Each such factor, like $x$ or $y-1$, vanishes along a line.

Figure 11. The line arrangements in the restricted argument. The integer labels record the powers with which the line factors occur in $P$; a negative power means division. Under the conditions just stated, intersecting lines have labels differing by one. The arrangements are two intersecting lines, or two parallel lines crossed by a third. The drawing shows which lines meet, not the whole complex space. Enlarge the line diagram.

The written argument concludes that a typical output has only one input in this class. It still needs independent review, and it does not cover arbitrary curved factors. There is no established reduction of every planar Keller map to this line case.

Why the limit condition cannot be dropped

Consider another rational comparison:

Its determinant is $1$. Even the differential check just discussed looks harmless: if $P=x^3y$ and $Q=1/(2x^2)$, then $P\,dQ=-y\,dx$, with no denominator left. The other expression, $Q\,dP$, also has polynomial coefficients.

Yet the map is still undefined at $x=0$. The distinct inputs $(t,t)$ and $(-t,-t)$ have the same output,

As positive $t$ shrinks toward zero, both inputs approach the origin while the shared output grows without bound.

Figure 12. A missing endpoint reveals the defect. At $t=1,1/2,1/4,1/8$, the second output is $1/2,2,8,32$. The collision pairs have a finite limit in the input space, but the formula is undefined there. That is exactly the failure the closedness condition detects. Enlarge the collision-limit diagram.

This comparison passes several necessary checks without being a polynomial map on the whole space. More complicated rational examples can hide their undefined points along curves. Examining selected lines, or even curves up to a fixed degree, can miss those defects. A proof must keep track of the complete domain as well as the identities.

Following loops and braiding inputs

Suppose an output travels around a loop. Follow each of its possible inputs continuously. When the output returns to its starting point, the inputs may have exchanged places. This behavior is called monodromy. A sheet is a local choice of one of those inputs.

Square roots make the idea visible. The equation $z^2=w$ has two roots when $w\ne0$. Let $w$ make one circuit around zero in the complex plane.

Figure 13. A loop can exchange the inputs. The root starting at $+1$ ends at $-1$ along the upper semicircle. The other travels from $-1$ to $+1$ along the lower semicircle. A second circuit returns both roots to their starting points. The square map has derivative zero at the origin, so this example illustrates sheet exchange without satisfying the conjecture’s hypothesis. Enlarge the loop diagram.

For several points moving in a plane, recording their paths over time gives a braid. A braid records how the paths wind around each other, including the order of their crossings. Counting how much each pair winds around one another does not retain all that information.

An exact experiment in the research notes illustrates the difference. A proposed pattern with six strands was adjusted until every pairwise winding count matched the required pattern. A full algebraic comparison still showed that the two braids were different. Later boundary checks also excluded that candidate.

This was a failed construction with a useful conclusion: getting all the numerical totals right does not necessarily produce the required geometry. The way the pieces fit together matters too.

What the OpenAI collection and recent papers contributed

The OpenAI mathematics collection was useful as a source of methods to examine. At the pinned 8 October snapshot, the review screened its catalogue of 719 manuscripts grouped into 372 families, then read selected statements, proofs, and formal-verification notes. It was not an audit of every proof.

The most useful outcome was a clearer understanding of the assumptions behind possible shortcuts:

• A theorem about compactified spaces requires connected fibers. A typical output with several separate inputs has a disconnected fiber, so that theorem cannot be applied directly to the possible counterexample under investigation.
• A result about how inputs exchange gives a restriction too weak, by itself, to exclude the possible exchanges among a finite number of inputs.
• The collection’s examples about removing a variable and changing coordinates after adding one examples warn that adding a variable can change an algebraic problem. Removing that variable requires a justification of its own.

Formal verification also has a precise scope. The collection’s logarithmic-additivity scope note identifies which part of that argument was formalized. A checked statement does not automatically certify every other statement in its paper, or a new argument built from it. None of the inspected results supplies the missing JC(2) step, and the calculations in this investigation must be assessed separately.

The arXiv review found useful results with similar limits. Shaska’s graded-map preprint proves invertibility for two-variable maps with a compatible scaling symmetry. A general map need not have that symmetry. Borisov, Gabber, and Vasiu give further conditional criteria, but the required conditions have not been established for every map.

Peretz’s October preprint studies escape on number grids with fixed denominators. Controlling those regularly spaced points does not control every possible path through complex space. The Kistner–Shaska constructions provide higher-dimensional examples, but no general way to turn them into a two-variable counterexample.

A random-tree approach offers another way to organize the inverse formula. An inverse can begin as an infinite series; it is polynomial only if all terms beyond some point vanish exactly. Showing that terms with large powers tend to have small coefficients is insufficient. Some reductions also add variables, so their conclusions need care when the question must remain in dimension two.

The accompanying X review encountered blocked posts and failed fetches. Their current contents were not verified. Search snippets and trend summaries were therefore not used as mathematical evidence. The review records the sources it could inspect; it does not claim to have found every recent result.

How to judge a claimed solution

For this problem, a few questions separate a promising idea from a completed answer:

• Is it about two complex variables? Higher dimensions and characteristic-two arithmetic address different cases.
• Are the formulas polynomial and defined everywhere? Division can hide excluded points or curves.
• Does the determinant identity hold at every input? Sample points and high-precision approximations cannot replace an exact identity.
• What does the argument actually cover? A counterexample needs one valid map that fails to be invertible. A proof needs to cover every map in the conjecture, not just a family with additional symmetry or a convenient form.

These questions apply to this investigation as much as to outside claims. A computer can verify a displayed identity or expose a failed example. That does not establish every general argument surrounding the calculation.

Where this update leaves the problem

The completed work has narrowed several proposed routes and made their missing steps more precise. The line-factor argument is one example of progress in a restricted setting; the rational comparisons explain why some apparently strong conditions are still insufficient.

There is no proof or counterexample to JC(2) in this article. The central question remains whether the polynomial condition in two complex variables prevents the escape that the verified three-variable map exhibits.

This article closes the research update through 9 October 2026. Further proof exploration is paused. The companion notes retain the full calculations for a later continuation.

The three-variable formula

This optional section gives the exact rule behind Figures 3–6. Let $s=1+xy$ be a shortcut to keep the expressions shorter, and define $F_3(x,y,z)=(A,B,C)$ by

Each line is one output coordinate. All three use only polynomial operations. The formal proof document, Sections 2–3 verifies $\det JF_3=-2$ everywhere and the three-point collision.

For the moving inputs in Figure 5, take $t\ge1$ and set

Substituting any of these into $F_3$ gives $(-1/(4t^2),0,0)$. At $t=1$, they are exactly the three inputs in Figure 3. The fractions here describe the paths of the inputs; they do not change the fact that $F_3$ itself is polynomial.

Sources and further reading

• Formal Verification of an Explicit Counterexample to the Jacobian Conjecture, Archive of Formal Proofs, July 2026. The machine-checked three-variable example and its higher-dimensional extensions.
• Shuhong Gao, “Counterexamples to the Jacobian conjecture in dimensions greater than two”, July 2026 preprint. The tangent-line explanation and further constructions.
• T. Shaska, “Graded Keller maps and the Jacobian Conjecture”, July 2026 preprint. A positive result for a specified family of two-variable maps.
• Romy Mondello, “A Dimension-Two Counterexample to the Separable Jacobian Conjecture in Characteristic Two”, July 2026 preprint. The result using different arithmetic.
• Guccione, Guccione, Horruitiner, and Valqui, “Increasing the degree of a possible counterexample to the Jacobian Conjecture from 100 to 108”, 2022 preprint. Restrictions on the polynomial degrees of a possible planar counterexample.
• Bass, Connell, and Wright, “The Jacobian conjecture: reduction of degree and formal expansion of the inverse”, 1982. A technical reference on inverse formulas and reductions that can change the number of variables.