How Fable AI Disproved the 87-Year-Old Jacobian Conjecture

A digestion of the Jacobian conjecture counterexample

I explore the surprising counterexample to the Jacobian conjecture recently discovered by Fable AI in three dimensions. While the conjecture remains open in two dimensions, this new finding reveals a polynomial map that is locally invertible yet globally non-invertible. I break down the geometric intuition behind this result, showing how simple polynomial multiplication and symmetry arguments explain a phenomenon that initially seemed like a mathematical miracle.

"The polynomial F has degree seven, so a priori the Jacobian ought to be a polynomial in three variables of degree as large as 18, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving 1299 coefficients."