What does this research mean for the field?

The Alpöge 2026 counterexample to the Jacobian conjecture in dimension 3 is computationally valid in exact rational arithmetic and can be extended into an infinite family of new counterexamples. Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The research aims to validate the Jacobian conjecture counterexample and extend its understanding through structural analysis and various geometric considerations.

July 28, 2026Open Access

The Jacobian counterexample, validated and extended: structure, an infinite family, exact escape geometry, and characteristic-p certificates

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The research aims to validate the Jacobian conjecture counterexample and extend its understanding through structural analysis and various geometric considerations.
Validated the counterexample using precise rational arithmetic.
Developed an infinite family of new counterexamples and explored exact escape geometry.
Established a rigidity theorem in dimension 2 and examined characteristic-p certificates.
Identified a unique mechanism validated by executed machine experiments.
Discovered multiple structural properties and counterexamples based on the findings.
Documented that the two-variable case of the conjecture remains unresolved.

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Implication

Foundational paper validates a counterexample in exact arithmetic and extends findings in algebraic geometry.

Cite This Study

Felipe Santibañez-Leal (2026) studied this question.

synapsesocial.com/papers/6a6850fb2845b684d16f01b1 https://doi.org/10.5281/zenodo.21579022

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The Jacobian counterexample, validated and extended: structure, an infinite family, exact escape geometry, and characteristic-p certificates

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The Jacobian counterexample, validated and extended: structure, an infinite family, exact escape geometry, and characteristic-p certificates

Key Points

Discussion

Implication

Cite This Study

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