The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.

Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse).

The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem, but is omitted here.) Also, from the fundamental theorem of algebra, once the Jacobian polynomial is non-zero, it must be constant. So the hypothesis “Jacobian is a non-zero constant” can be replaced with “ is locally invertible”. So the Jacobian conjecture can be viewed as an assertion that local invertibility implies global invertibility. The complex numbers can be easily replaced with other fields of characteristic zero by the Lefschetz principle, but I prefer to work in the concrete setting of the complex numbers.

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):

Theorem 2 (Counterexample to conjecture) There exists a polynomial which has non-zero constant Jacobian, but is not invertible.

The conjecture remains open in two dimensions, and is easy to establish in one dimension.

The example can be stated completely explicitly: one can take

and one can verify by a brief calculation that

and

While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial

has degree seven, so

a priori

the Jacobian

ought to be a polynomial in three variables of degree as large as

, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving

equations, which is much larger than the

degrees of freedom for a generic degree seven polynomial of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.

The example has since been retroactively explained in more geometric terms. As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places were some remarkable phenomena occur.

It is convenient to use the local injectivity formulation, and to generalize the domain to an equivalent affine variety. Namely, we will show

Theorem 3 (Counterexample, reformulated) There exists an affine variety that is isomorphic to by polynomial changes of variable, and a polynomial map which is locally injective, but not globally injective.

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian. Our objective is now to find data , that obeys three separate properties:

• (a) is locally injective on .
• (b) is not globally injective on .
• (c) is isomorphic to by polynomial changes of variable.

The advantage of splitting the problem in to these three components is that we can build towards each of them separately.

It turns out that and can be built out of the operation of multiplication of low degree polynomials. Namely, consider the following three simple affine spaces:

• The space of linear homogeneous polynomials of two complex variables .
• The space of quadratic homogeneous polynomials of two complex variables .
• The space of cubic homogeneous polynomials of two complex variables .

(The notation

here refers to the

symmetric power

of a vector space

.) Clearly these spaces are isomorphic to

respectively. Furthermore, we have a multiplication map

, mapping a pair

of a linear polynomial

and a quadratic polynomial

to a cubic polynomial

(Right now, the domain and range of this map

is larger dimensional than the target of three; we will cut the dimensions down to three as the argument progresses.)

The map , essentially a map from to , is clearly polynomial; it is given explicitly in coordinates as

The map

also enjoys two basic (and commuting) symmetries:

• If one applies a scaling for some non-zero complex numbers , then the product is scaled by : .
• If one applies a change of variables for some invertible linear transformation , then the product is transformed by : .

So this map enjoys a huge amount of equivariance, basically with respect to an action of the five-dimensional group

.

The five-dimensional domain is of course larger than the four-dimensional range , so the map clearly cannot be injective. This can already be seen from the scaling symmetry, as the specific scalings

for

modify the linear and quadratic polynomials

but not their product

. But even if one quotients out by this symmetry

(3)

to cut the dimension of the domain down to four, the map

is still not injective for the following basic reason. A generically chosen cubic polynomial

will split into the product

of three independent linear polynomials. Then there are three pairs

which all map to the same cubic polynomial

under the multiplication map

, but are not related to each other by scaling symmetry

(3)

. Thus, we see that even after quotienting out by the scaling symmetry

(3)

, the multiplication map

is generically non-injective in a three-to-one fashion. Thus we already have achieved something resembling goal (b)!

It will be convenient to “spend” the scaling symmetry to obtain a useful normalization. If is a linear polynomial and is a quadratic polynomial, the resultant can be defined by the determinant

If we have a factoring

then the resultant can also be described as

Thus the resultant measures whether the linear polynomial

and the quadratic polynomial

share a common root. A fundamental fact about resultants is that they are

-invariant: for any

, we have

One way to see this is to check it first for translations

(which translate the roots

by

while leaving

unchanged) and for inversions

(which map

to

while mapping

to

and

respectively), and then noting that these transformations generate all of

. They also interact very nicely with scaling:

In particular, the scaling symmetry

(3)

multiplies

by

:

Thus, we can (generically) normalize away this scaling symmetry by imposing the condition

We now have a restricted multiplication map (which by abuse of notation we will continue to call ) from the four-dimensional variety

to the four-dimensional space

. This map

is still not globally injective, as we can take the three pairs in

(4)

from before and apply the scaling

(3)

separately to each of the three pairs to obtain the normalization

(7)

. So we have kept property (b). Furthermore, this map retains the

-equivariance (and also one remaining scaling symmetry, though we will not make much further use of that symmetry).

But we now also have property (a)! Suppose we want to show the local injectivity of in the neighborhood of a pair with . As the resultant is non-vanishing, the root of (which exists in the Riemann sphere, or projective line if you prefer) is distinct from the two roots of (though the latter two roots could be equal to each other). Applying the action (which performs Möbius transforms on the roots), one can assume without loss of generality that is the point at infinity (or equivalently ), thus for some complex number and for some complex numbers , with the resultant condition (7) simplifies to (so in particular are also non-zero). It is then clear that if one perturbs and by a small amount (say, modifying each coefficient by ), then the root of will perturb to something large (), while the roots of stay bounded. Thus, just from knowledge of the product , one can reconstruct which of the three roots of this cubic polynomial will be the perturbed root of , and which two will be the perturbed roots of ; from this and (6), (7) we can also reconstruct the leading coefficient of , and this completely determines both and . This establishes the local injectivity property (a). (In fact it is étale, but we will not need the machinery of étale maps here.)

Unfortunately, (the four-dimensional analogue of) condition (c) fails: the quadric hypersurface (8) is not isomorphic to the affine space . But we can try to get around this by passing to a three-dimensional slice. Let be some three-dimensional affine plane of (which we will take to avoid the origin for technical reasons), then we can restrict as a map from the set

to

. The latter is clearly identifiable (by linear changes of coordinate) to

. As

was already locally invertible, it remains locally invertible under restriction; and because generic cubic polynomials

had three preimages under

in

(8)

, this continues to be the case after restricting to

(9)

(unless

was somehow so degenerate that it had no generic elements, but this turns out to be impossible). So we have retained properties (a) and (b). The miracle is that, with a good choice of

, we can also obtain (c) and obtain the desired counterexample to the Jacobian conjecture: despite appearances, the variety

(9)

is in fact equivalent to the affine space

by polynomial changes of variable!

Let’s see how. The affine hyperplanes in avoiding the origin are parameterized by the dual space of avoiding the origin, which one can think of as the non-zero third order homogeneous differential operators in two variables. Indeed, every such operator generates an affine hyperplane that avoids the origin, and conversely by duality every affine hyperplane avoiding the origin arises in this form uniquely. Just as the cubic polynomials in can be factored into three linear polynomials, the differential operators in the dual space can also be factored into three linear differential operators, e.g.,

in the case that

is non-zero. The

action moves the roots

around the Riemann sphere by Möbius transformations. As these transformations are

-transitive, the actual selection of such roots is not too important (and the scaling symmetry similarly makes the choice of leading coefficient

unimportant); the only thing to keep track of is whether the roots repeat. Up to the symmetries, there are in fact just three different equivalence classes of differential operator

(and thus of affine hyperplane

) to consider:

• Operators where the three roots are all distinct, thus for independent first-order operators .
• Operators where two roots coincide and one is distinct, thus for independent first-order operators .
• Operators where all three roots coincide, thus for some first-order operator .

It turns out that the affine miracle for (9) occurs precisely in the second case, when has two identical roots. I do not have a completely satisfactory geometric explanation for this miracle, but one can verify it by the following coordinate computation.

By applying the action, we can normalize so that , thus is now the affine hyperplane of cubic polynomials with . Using (2) and (5), the variety (9) can now be described explicitly in coordinates as

At first glance this seems to be a generic-looking variety cut out by a cubic equation and a quadratic equation – hardly a candidate to be affine! But observe that if

is non-zero, then the second equation

can be solved for

,

and the first equation

can be solved for

,

Putting these two equations together, we see that as long as one removes the case

, the quintuple

is uniquely determined by

by a change of variables which is Laurent in

and polynomial in

. Thus we have a nice birational equivalence

Thus we have already

almost

established property (c): the variety

(9)

becomes birationally equivalent to

after cutting out the

subvariety. In particular, for each fixed non-zero value

of

, the corresponding fiber

of

(10)

is equivalent to

by polynomial changes of variable, since we can reconstruct

from the coordinates

by the polynomial formulae

So we just need to glue back in the fiber. Indeed, from (10) we see that the fiber at is just

Now we observe a key miracle: the cubic equation

and quadratic equation

have a unique affine solution

(as opposed to the six possible solutions that Bezout’s theorem might suggest – the other five solutions live on the line at infinity). So the fiber here is also affine:

This is extremely encouraging for the purposes of establishing property (c), as it strongly suggests that the variety

(10)

has the structure of an

-bundle over

, which is already extremely close to being isomorphic to the affine space

. The main remaining task is to make sure that nothing singular happens in the limit

, and that a global polynomial coordinate chart for

(10)

that covers both the

and

fibers can be constructed.

The standard way to proceed here is to manipulate various tangent spaces using the modern machinery of algebraic geometry and commutative algebra, but given my own background, I prefer to adopt the language of analysis, and in particular big-O notation (in place of the ideals used in algebraic geometry), in order to investigate the limit by hand. On the variety (10), let us use to denote any multiple of by a polynomial expression in . Thus, for instance, the equation implies that

while the equation

implies that

as well as the more refined estimate

In the

case we could conclude that

. Now we perturb this observation. Multiplying

(13)

by

we have

, which on substitution into

(14)

gives

; substituting this back into either

(13)

or

(14)

also gives

.

We can get some more precise asymptotics by also taking advantage of (15). Substituting into (15), we obtain after some algebra

So if we write

more explicitly as

, then we have

and thus

Substituting this back into

(11)

gives an asymptotic for

:

Finally, one can insert these estimates into

(12)

, although one only gets a trivial bound in this case:

Expanding the error term in (16) as , and doing a little more algebra, we thus have a polynomial change of variables

which completely parameterizes the variety

(10)

by polynomial combinations of three coordinates

. This already gives (a) and thus completes the proof of Theorem

3

.

The previous computations, when expanded out, also gives polynomial inverse maps:

The map from

to the

coefficients of

(dropping the

coefficient which is constrained to equal

), we obtain a polynomial map

with

which theory predicts to have a constant Jacobian, and indeed one can calculate that the Jacobian is

. This is essentially the original example up to trivial changes of variable; indeed, one can check that the map

is exactly the map

given in

(1)

.

AI disclosure: I used an AI chatbot to discuss various aspects of this problem and to confirm several of the calculations made here.