Jacobian Conjecture for Baby

This bending can be perfectly undone.

Look at this animation. A flat grid gets bent into a strange curved shape. It looks like the grid is ruined.

It is not. This bending can be perfectly undone. There is a second simple rule that moves every point back to exactly where it started. Nothing is lost.

In 1939, a mathematician named Ott-Heinrich Keller asked a simple question about maps like this one. His question became one of the most famous unsolved problems in mathematics. Many mathematicians published proofs. Every proof turned out to be wrong.

The problem stayed open for 87 years. Then, in July 2026, it fell. Only partly, and in a way nobody expected. (The original two-dimensional case is still open.)

This article explains the whole story from zero. You do not need any math background. Each chapter is one small idea, explained with a picture first. At the end, you can check the 2026 discovery with your own hands , using simple arithmetic.

(Everything here is backed by an open repository: github.com/Muchmirul/jacobian-conjecture. It contains the code for every animation and a test suite that re-checks every mathematical claim.)

Here is the plan. Chapters 1 to 8 build the ideas one by one. Chapter 9 states the famous question. Chapters 10 to 12 tell what happened to it.

1 · Functions are machines

We start with the smallest idea: what a “function” is.

A function is a machine. You feed it a number. It gives you back a number. That is all.

One number at a time: 3 to 7, 0 to 1, 10 to 21.

The machine above uses the rule “double the number, then add 1.” Feed it 3, you get 7. Feed it 0, you get 1. Feed it 10, you get 21.

Two rules make a machine a function. First, it always gives an answer. Second, the same input always gives the same output, with no randomness and no “it depends”.

Now change the viewpoint. This new picture powers the rest of the article. Instead of feeding the machine one number at a time, imagine it moving every number on the number line at the same time .

Same machine, new picture: a function moves the whole line. Keep this picture in mind. We will soon use it on bigger spaces.

One piece of notation, and it is the only one we need. Mathematicians name a machine with a letter like $f$ and write its rule as

$$ f(x) = 2x + 1. $$

Read it as “$f$ takes a number $x$ and returns $2x + 1$.” The letter $x$ is just a placeholder for the input.

The one thing to remember: a function is a machine that turns each input into exactly one output, and you can picture it moving every number at once.

2 · The undo machine

A machine moves every number. The next question is whether the move can be reversed. This small idea is the seed of the whole story.

The machine “double it” sends 3 to 6. The machine “halve it” sends 6 back to 3. Run one after the other, and every number ends up exactly where it started.

When a second machine brings every output back to its original input, it is called the inverse , or the undo machine. Doubling and halving undo each other. So do “add 5” and “subtract 5”.

Now try to undo “square it”, which multiplies the number by itself.

Left: −3 and 3 land on the same output, so undo is impossible. Right: no two arrows meet, so every arrow can be traced back.

Feed it −3: you get 9. Feed it 3: you also get 9. Now stand at 9 and try to walk back. Which input did you come from? You cannot know. The information “was it negative?” is gone.

An undo machine cannot exist when either of two things happens:

Collisions: two different inputs land on the same output. (−3 and 3 both land on 9.)

Gaps: some outputs are never produced, so the undo machine would have nothing to say there. (Squaring never outputs −4.)

If there are no collisions and no gaps, then every output has exactly one origin. So “walk back where you came from” is itself a machine.

(The official words, if you want them: no collisions is called injective , no gaps is called surjective , both together is called invertible . You can forget these words. “No collisions, no gaps” is all you need.)

The one thing to remember: a machine can be undone exactly when no two inputs share an output and no output is missing. Undoing fails the moment information is destroyed.

3 · Polynomials

So far a machine could use any rule. The famous problem is about one specific family of machines: the ones built from nothing but plus and times.

Take an input $x$. Allow yourself only two operations, adding and multiplying, using numbers or pieces you have already built. Any machine you can build this way is called a polynomial .

$2x + 1$ doubles and then adds one. It is a polynomial.

$x^2$ is $x$ times $x$. It is a polynomial, since a power is just repeated multiplication.

$x^3 - 2x$ builds $x \cdot x \cdot x$, builds $2x$, and subtracts. It is a polynomial, since subtracting is adding a negative.

Here are their graphs. For each input $x$ along the bottom, the height of the curve shows the...