Graded Keller maps and the Jacobian Conjecture

Abstract

Rational points on weighted projective spaces are sparse: a point of $\mathbb{P}^n(\mathbb{Q})$ lifts along the Veronese morphism only when a Kummer condition holds at every prime (arXiv:2509.02319). We ask what this viewpoint says about the Jacobian Conjecture. The map recently announced as a counterexample is equivariant for the grading $\mathrm{wt}(x,y,z)=(1,-1,-2)$, and we show that the sign pattern of such a grading decides everything. If the weights are all positive, the setting of weighted projective spaces, an equivariant Keller map is always an automorphism, so no counterexample can be graded that way. In dimension two the same holds for every sign pattern. The Keller condition itself descends to the quotient, where it says that the Jacobian of the quotient map vanishes to order two along the contracted locus. Over $\mathbb{Q}$ the image of the counterexample is a thin set. But along the line where the group action has stabilizer $\mu_2$, the two preimages on the contracted locus are rational exactly when $-a$ is a square, so the Kummer condition of the positive-weight theory reappears on the stacky stratum. We propose a classification of equivariant polynomial maps by the signature of the weights, with weighted projective geometry as the positive case.