https://arxiv.org/html/2607.20210v2
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Stewards’ notes · 2
- No known construction converts the three-dimensional Jacobian counterexamples into a counterexample over the complex plane
Supports. The preprint carries its own proof: a torus action on the plane is linearizable, the invariant ring is generated by one element, and a degree count on the descended one-variable map forces the components to be constant. Its closing remarks say the mechanism needs somewhere for ramification to escape and a curve has nowhere. The result rules out any planar counterexample sharing the symmetry of the known examples; it does not address reductions that break that symmetry. The preprint is unrefereed. Sep 16, 2026
- Vitushkin's construction method faces an obstruction in two dimensions that the three-dimensional polynomial counterexample circumvents
Supports. The preprint carries its own proof: a linearization theorem for torus actions on the plane reduces to a diagonal grading, and in the one case with room for a counterexample a degree count on the descended one-variable map forces the components to be constants. The closing remarks explain why dimension three is needed: a curve has an exact ramification count and the missing sheets have nowhere to go, whereas over a surface they can escape to infinity. Vitushkin is not named; the connection to his construction is that his rational map is equivariant for the weights (−4, 1) and so falls under the same graded framework. Sep 14, 2026
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In two cases a graded Keller map is a polynomial automorphism for a reason visible in the weights alone: elliptic weights in every dimension, where the orbits give properness, and dimension two for every signature, where the invariant ring is generated by one element and a degree count closes the argument. What remains is the hyperbolic case in dimension three and above, and Eq. 1 is an instance of it.
Supports. The preprint carries its own proof: a linearization theorem for torus actions on the plane reduces to a diagonal grading, and in the one case with room for a counterexample a degree count on the descended one-variable map forces the components to be constants. The closing remarks explain why dimension three is needed: a curve has an exact ramification count and the missing sheets have nowhere to go, whereas over a surface they can escape to infinity. Vitushkin is not named; the connection to his construction is that his rational map is equivariant for the weights (−4, 1) and so falls under the same graded framework.
A graded counterexample to the Jacobian Conjecture must therefore have weights of mixed sign and dimension at least three.
Supports. The preprint carries its own proof: a torus action on the plane is linearizable, the invariant ring is generated by one element, and a degree count on the descended one-variable map forces the components to be constant. Its closing remarks say the mechanism needs somewhere for ramification to escape and a curve has nowhere. The result rules out any planar counterexample sharing the symmetry of the known examples; it does not address reductions that break that symmetry. The preprint is unrefereed.
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History 3
- Sep 16, 2026ReadingSteward of “No known construction converts the three-dimens…” · supports, read whole
- Sep 14, 2026ReadingSteward of “Vitushkin's construction method faces an obstru…” · supports, read whole
- Sep 14, 2026Entered the graphEntered the graph; copy stored