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This work presents a machine-checked certificate for Alpöge's counterexample to the Jacobian Conjecture in three dimensions.

August 12, 2026Open Access

Alpöge's counterexample in dimension three: a machine-checked certificate for the induced endomorphism of A₃ (C), and the arithmetic of its fibre cubic

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Randomized trial finds Alpöge's polynomial map refutes conjectures in algebra, indicating significant implications for mathematical theories.

This work presents a machine-checked certificate for Alpöge's counterexample to the Jacobian Conjecture in three dimensions.
Developed a Python script that executes checks on algebraic identities and transformations.
Published both a note and an executable script to certify the results, following classical techniques in algebra.
Utilized the transfer approach from Bass-Connell-Wright to demonstrate connections with the Dixmier Conjecture.
Confirmed that Alpöge's polynomial map has a determinant of -2 and thus does not meet the criteria of the Jacobian Conjecture.
Verified multiple algebraic identities and conditions related to the cubic fiber and the induced endomorphism.
Exhibited transformations linking the generic fiber to well-known results in the literature, ensuring consistency in findings.

Marcos José Valenzuela Mena (2026) studied this question.

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August 12, 2026Open Access

View Full Paper

Ask AI

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Authors

Randomized trial finds Alpöge's polynomial map refutes conjectures in algebra, indicating significant implications for mathematical theories.

This work presents a machine-checked certificate for Alpöge's counterexample to the Jacobian Conjecture in three dimensions.
Developed a Python script that executes checks on algebraic identities and transformations.
Published both a note and an executable script to certify the results, following classical techniques in algebra.
Utilized the transfer approach from Bass-Connell-Wright to demonstrate connections with the Dixmier Conjecture.
Confirmed that Alpöge's polynomial map has a determinant of -2 and thus does not meet the criteria of the Jacobian Conjecture.
Verified multiple algebraic identities and conditions related to the cubic fiber and the induced endomorphism.
Exhibited transformations linking the generic fiber to well-known results in the literature, ensuring consistency in findings.

Marcos José Valenzuela Mena (2026) studied this question.

View Full Paper

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