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August 2, 2026Open Access

Binary forms, discriminants and Keller maps in dimension three

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Authors

OAOzorio Olea Arnaldo Adrian

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Overview

Counterexample demonstrates failure of the Jacobian Conjecture in three dimensions, implying new geometric insights.

Key Points

  • The central aim is to construct a counterexample to the Jacobian Conjecture in three dimensions using a novel geometric framework.
  • Constructed a Quasi-Homogeneous Invariant Projection Space (QH-IPS) linked to a Keller map F: C³ → C³.
  • Defined a projective fibration I → C³ over the target identified with roots of a binary form related to F.
  • Studied structural identities and discriminants to analyze map behavior in the context of integral geometry.
  • Created a Keller map with a constant Jacobian of -2, mapping three distinct points to the same target value.
  • Identified the source C³ as the open set of simple roots of a binary cubic form, with three colliding roots for a specific target.
  • Demonstrated that for arbitrary degree d ≥ 3, the construction allows infinite Keller maps, with the Galois group being the full symmetric group S_d for d ≥ 4.

Cite This Study

Ozorio Olea Arnaldo Adrian (2026) studied this question.

synapsesocial.com/papers/6a6eeb011b0468a7eeab3c47https://doi.org/10.5281/zenodo.21696160
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