Jacobian Conjecture for Baby
9 hours ago
- A function is a machine that maps each input to exactly one output, and can be visualized as moving every point on a number line.
- A function is undoable if there are no collisions (two inputs to the same output) and no gaps (outputs without an input), making it invertible.
- Polynomials are machines built only from addition and multiplication, the simplest concretely checkable functions.
- A plane map moves every point of the plane using two polynomial rules; it can be undone if its Jacobian determinant (local area factor) is nonzero everywhere.
- Local undoability (nonzero Jacobian everywhere) does not guarantee global undoability; collisions can occur between distant points unseen by a local microscope.
- The Jacobian Conjecture asks: if a polynomial map's local area factor is a constant nonzero everywhere, must it be globally undoable with a polynomial inverse?
- The conjecture was unsolved for 87 years, with all tests confirming it, partial proofs for low degrees, and obstacles including real-number counterexamples and clock arithmetic failures.
- In July 2026, an explicit degree-7 counterexample in three dimensions was discovered: a polynomial map with constant Jacobian determinant -2 but three different points mapping to the same output, disproving the conjecture for dimensions 3 and higher, while the original two-dimensional case remains open.