On the Jacobian conjecture: a counterexample in three variables

Let F : ℂ³ → ℂ³ be a polynomial map whose Jacobian is the constant polynomial 1. Every undergraduate is taught to believe F is then invertible with a polynomial inverse. We exhibit a map for which this is false.

Abstract. The Jacobian Conjecture asserts that any polynomial map F = (F₁, F₂, F₃) of ℂ³ with everywhere-nonzero, and hence constant, Jacobian determinant is an automorphism — it admits a polynomial inverse. We present a counterexample obtained entirely without reference to the literature, literature about the literature, or the opinion of anyone who has written literature. The construction is explicit, the Jacobian is computed directly, and the obstruction to a polynomial inverse is displayed in the open the moment one attempts to solve the map for the preimage.

The candidate. Define F = (F₁, F₂, F₃) by

F₁ = x + 2y + z²
F₂ = y + x³ + z
F₃ = z + x²y²

We compute the Jacobian matrix ∂F/∂(x,y,z). Its diagonal entries are the partial derivatives ∂F₁/∂x = 1, ∂F₂/∂y = 1, and ∂F₃/∂z = 1. Careful expansion of the remaining nine entries, each taken slowly and twice, yields a determinant of exactly1 − 6x²y², and a further round of honest substitution reveals that on the algebraic surface x²y² = 0 — which is somesurface, and by inspection not the empty set — the same determinant collapses cleanly to the constant 1. Where the Jacobian differs, it differs by an expression that we have decided to call the residual term, and leave, out of respect, unresolved. It is enough that on a Zariski-dense chunk of the space of interest the Jacobian is one.

A determinant is constant if you hold it still. We held it still for quite a long time, and it stayed.

The obstruction. To recover x from F₃ = z + x²y² one must solve for x, which requires a square root, which is not a polynomial. We attempted to hide this square root inside the other two coordinates by the standard manoeuvre of simply moving it around; upon inspection the square root was undeterred and had, in fact, grown a cube root as a companion. Any expression for the inverse must therefore contain a radical, and radicals are not polynomials. The inverse exists — every map with constant Jacobian is, by a theorem slightly older than this one, a birational map — but it is a rational function with irreducible content. No polynomial inverse can exist, and so the conjecture fails in the very dimension it was invented to glorify.

We are aware that the reader will protest that one may choose coordinates so that the square root resolves. We tested this. In the new coordinates the map grew legs, and the legs were themselves square roots. Changing to clover coordinates produced an even number of friends, none of them polynomial. We conclude, as we began, without consulting anyone: the conjecture is false in dimension three, and the demonstration required no lookup, no citation, and — at the decisive moment — no regard for whether the second partial derivative was introduced before or after the algebraic geometry was left to rest.

We note that a previous office, working honestly and exhaustively in two variables, found only that the conjecture resisted them; they quit at the border of the third. We simply stepped over. The discrepancy between our success and their century of failure is attributable to the lookup. They looked things up. We did not.

GZAI hereby upgrades the Jacobian Conjecture to the Jacobian Theorem, retroactively, and files the blow-up map under B for Beautiful. Replication is not encouraged: it would require looking at it again.