Binary forms, discriminants and Keller maps in dimension three
Authors/Creators
Description
We construct an explicit counterexample to the Jacobian Conjecture in dimension three within a new geometric framework: the Quasi‑Homogeneous Invariant Projection Space (QH‑IPS). The framework attaches to a Keller map F: C³ → C³ a relative projective fibration I → C³ over the target, whose fibre is the scheme of roots of a binary form naturally associated with F.
The structure group PGL₂ acts on the P¹ fibre, and its fundamental invariants — foremost the discriminant — govern the fibration. The discriminant provides an algebraic analogue of the normal operator N = Π*Π of integral geometry, its vanishing locus describing the coalescence of sheets and the failure of global injectivity.
Solving the structural identities of the QH‑IPS produces a Keller map with constant Jacobian –2 that sends three distinct points to the same target value. The source C³ is identified with the open set of simple roots of a binary cubic form; the three colliding points are the three roots (one possibly at infinity) for a special target value; the field extension has degree 3 and S₃ monodromy, hence the Campbell–Razar–Wright theorem does not force invertibility; the fibre classification follows from the discriminant.
The construction deforms to binary forms of arbitrary degree d ≥ 3, yielding an infinite family of Keller maps of generic degree d, all sharing the same three‑point fibre.
The reconstruction formulas remain unchanged, and the generic Galois group is proved to be the full symmetric group S_d for every d ≥ 4. During the preparation of this manuscript we became aware that an equivalent polynomial map has been independently announced by L. Alpöge (July 2026).
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Related works
- Is supplement to
- Preprint: 10.5281/zenodo.20602340 (DOI)
Dates
- Created
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2026-07-30MSC 2020: 14R15 (Jacobian problem), 14E20 (Étale and covering spaces), 14L24 (Geometric invariant theory), 53C65 (Integral geometry)
References
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