Endomorphisms of affine spaces and the Jacobian problem
arxiv.org
arxiv.org
And with this paper, it looks like it may have come from an unpublished but publically available draft:
> Supplemental AI statement of the third author. Very unfortunately and without me being aware, the in-progress version of this work dated January 14, 2025 was shared publicly at the link https://www.ihes.fr/~gabber/BGV160.pdf on January 16, 2025. My understanding is that after 18 months it became a feed of the AI models. On July 23, a graduate student sent me the 37 pages pdf file at the link https://people.math.binghamton.edu/adrian/AI.pdf produced by AI when prompted on how the 7 pages AI paper https://www.ulam.ai/research/jacobian.pdf relates to BGV160.pdf; a typo in the statement of Theorem 30.1(2.j) was fixed based on what the file mentioned (its application to Theorem 30.1(10.c) did not require any change).
> To my best knowledge:
>(1) there has been no prior work before BGV160.pdf aiming to classify all étale endomorphisms of the affine spaces that have geometric degrees 3, see Section 26 of BGV160.pdf
> (2) there has been no prior work before BGV160.pdf that used systematically xy, 1+xy, and δ +xy in the construction of open embeddings, morphisms, and étale endomorphisms, including the morphism A2 K → P1 K defined by the rule (x : 1 + xy);
> (3) between BGV160.pdf and the 7 pages AI paper there exists overlap of notation (such as Γ for complements) and of intent (such as the variation of the geometric degrees, counting fibers, non-proper locus, families, the usage of 1 + xy, the computation of the normalization Xe which actually plays no role in terms of the counterexamples to the Jacobian Conjecture, etc.) which are also very pertinently and systematically presented in the 37 pages pdf file;
> (4) as of today the prompt history, the model version, and the workflow behind the 7 pages AI paper have not been disclosed publicly.