Dimensions, reductions, and consequences

Every dimension at least three

For \(n>3\), define

\[ F^{(n)}(x,y,z,t_4,\ldots,t_n) =(F_1(x,y,z),F_2(x,y,z),F_3(x,y,z),t_4,\ldots,t_n). \]

Its Jacobian matrix is block diagonal, so

\[ \det JF^{(n)}=-2. \]

Padding the three colliding points with the same extra coordinates preserves the collision. Hence the conjecture is false in every dimension \(n\ge3\).

Dimension two remains separate

Nothing here projects to a two-variable Keller counterexample. As immediately pre-announcement literature records, even \(n=2\) remained unresolved [Bisi et al., 2026, Makar-Limanov, 2025, Rodríguez Díaz, 2026]. A counterexample in dimension three settles the unrestricted all-dimensional statement negatively, but does not settle the plane problem.

Compatibility with classical reductions

Bass–Connell–Wright and Yagzhev reduce the all-dimensional conjecture to maps of the form identity plus a cubic homogeneous part, after adding variables [Bass et al., 1982]. Drużkowski gives a further cubic-linear reduction [Drużkowski, 1983], and de Bondt–van den Essen give a symmetric reduction [de Bondt and van den Essen, 2005].

Therefore this example implies the existence of counterexamples in those reduced forms in some larger dimensions. It does not imply the existence of a cubic counterexample in dimension three. There is no conflict with Wang’s degree-at-most-two theorem [Wang, 1980] or special cubic results in dimension three [Wright, 1993].

Dixmier conjecture

The standard rank-preserving implication \(DC_n\Rightarrow JC_n\) sends a Keller map to a Weyl algebra endomorphism. For the normalized map \(H\), let

\[ \delta_i=\sum_j (JH^{-1})_{ji}\,\partial_j. \]

Since \(\det JH=1\), every coefficient of \(JH^{-1}\) is polynomial. The identities

\[ \delta_i(H_j)=\delta_{ij},\qquad [\delta_i,\delta_j]=0 \]

show that

\[ X_i\longmapsto H_i,\qquad \partial_i\longmapsto\delta_i \]

defines an endomorphism of the third Weyl algebra. The included SymPy certificate checks these relations exactly. Were this endomorphism an automorphism, the standard centralizer argument would force every \(X_j\) to lie in \(\mathbb C[H_1,H_2,H_3]\), giving a polynomial inverse for \(H\) and contradicting its three-point fiber. It is therefore an explicit non-automorphic Weyl-algebra endomorphism: the rank-three Dixmier conjecture is false.

Belov-Kanel–Kontsevich and Tsuchimoto proved the opposite indexed implication \(JC_{2n}\Rightarrow DC_n\); together with the older \(DC_n\Rightarrow JC_n\), this yields stable equivalence [Belov-Kanel and Kontsevich, 2007, Tsuchimoto, 2005].

This conclusion should be kept rank-specific: it does not by itself settle the rank-one or rank-two Dixmier conjectures.

Field scope

The same polynomial and collision work over any field of characteristic not equal to \(2\). In odd positive characteristic this is another counterexample, but the positive-characteristic analogue was already false for elementary Frobenius reasons. The historical breakthrough is over characteristic zero.