Graded Keller maps and the Jacobian Conjecture

Tanush Shaska

I-AIMS preprint sh-131 · 2026

We study graded Keller maps: polynomial maps ℂⁿ → ℂⁿ equivariant for linear actions of 𝔾ₘ on the source and on the target, with source weights w = (w₁, …, wₙ). We ask what the signature of w decides. If the wᵢ are nonzero and of one sign, or if n ≤ 2, then a graded Keller map is a polynomial automorphism. A graded counterexample to the Jacobian Conjecture must therefore have weights of mixed sign and n ≥ 3, and the map recently announced as a counterexample is graded with w = (1, −1, −2). For w = (1, −r₂, …, −rₙ) with all rᵢ > 0 the Keller condition descends to the invariant quotient 𝔸ⁿ⁻¹, where it says that the Jacobian of the quotient map is a nonzero constant times the (m−1)st power of the equation of the contracted locus, m = r₂ + ⋯ + rₙ. That identity is multilinear in its unknowns, so the graded counterexamples of bounded degree are exactly the points of an explicit multiprojective scheme over ℤ, one of whose rational points is the announced map. Modulo the natural 𝔾ₘ the parameter space is a weighted projective space with positive weights, so a hyperbolic counterexample is parametrized by an elliptic moduli space, on which rational points satisfy a Kummer condition at every prime. We classify graded Keller maps by the signature of w: completely for elliptic weights, by reduction to the ungraded conjecture in the number of zero weights for parabolic weights, and by reduction to that scheme for hyperbolic weights. Over ℚ the image of the announced map is thin, but along the line where the stabilizer is μ₂ its two preimages on the contracted locus are rational exactly when -a is a square.

MSC classes: Primary 14R15; Secondary 14R10, 14R20, 14G05, 14M25

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