References

References#

[Alpoge26]

Levent Alpöge. X post announcing the explicit three-dimensional polynomial map. X post, 2026. Posted 20 July 2026 at 02:19:17 UTC; original public announcement of the displayed formula. URL: https://x.com/__alpoge__/status/2079028340955197566.

[Bar22]

Wolfgang Bartenwerfer. The cremona problem in dimension 2. Archiv der Mathematik, 119:53–62, 2022. URL: https://link.springer.com/article/10.1007/s00013-022-01733-1, doi:10.1007/s00013-022-01733-1.

[BCW82]

Hyman Bass, Edwin H. Connell, and David Wright. The jacobian conjecture: reduction of degree and formal expansion of the inverse. Bulletin of the American Mathematical Society, 7(2):287–330, 1982. URL: https://www.ams.org/journals/bull/1982-07-02/S0273-0979-1982-15032-7/S0273-0979-1982-15032-7.pdf, doi:10.1090/S0273-0979-1982-15032-7.

[BKK07]

Alexei Belov-Kanel and Maxim Kontsevich. The jacobian conjecture is stably equivalent to the dixmier conjecture. Moscow Mathematical Journal, 7(2):209–218, 2007. URL: https://arxiv.org/abs/math/0512171, doi:10.17323/1609-4514-2007-7-2-209-218.

[BV17]

Angelo Calil Bianchi and Marcelo Oliveira Veloso. Locally nilpotent derivations and automorphism groups of certain danielewski surfaces. Journal of Algebra, 469:96–108, 2017. URL: https://arxiv.org/abs/1506.00164, doi:10.1016/j.jalgebra.2016.08.030.

[BDG+26]

Elia Bisi, Piotr Dyszewski, Nina Gantert, Samuel G. G. Johnston, Joscha Prochno, and Dominik Schmid. Random planar trees and the jacobian conjecture. Journal of the London Mathematical Society, 113(1):e70416, 2026. URL: https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.70416, doi:10.1112/jlms.70416.

[Bor20]

Alexander Borisov. Frameworks for two-dimensional keller maps. The Electronic Journal of Combinatorics, 27(3):P3.54, 2020. URL: https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p54/pdf/, doi:10.37236/9210.

[BGV26]

Alexander Borisov, Ofer Gabber, and Adrian Vasiu. On endomorphisms of affine spaces and the jacobian problem. 2026. Current manuscript, accessed 20 July 2026. URL: https://www.ihes.fr/%7Egabber/BGV160.pdf.

[BKrasinski23]

Szymon Brzostowski and Tadeusz Krasiński. A note on the paper “the cremona problem in dimension 2”. 2023. URL: https://arxiv.org/abs/2306.03996, arXiv:2306.03996.

[Cam73]

L. Andrew Campbell. A condition for a polynomial map to be invertible. Mathematische Annalen, 205:243–248, 1973. URL: https://doi.org/10.1007/BF01349234, doi:10.1007/BF01349234.

[Cha99]

Nguyen Van Chau. Non-zero constant jacobian polynomial maps of ℂ². Annales Polonici Mathematici, 71(3):287–310, 1999. URL: https://matwbn.icm.edu.pl/ksiazki/apm/apm71/apm7135.pdf.

[Cha04]

Nguyen Van Chau. Note on the jacobian condition and the non-proper value set. Annales Polonici Mathematici, 84:203–210, 2004. URL: https://arxiv.org/abs/math/0305088.

[CN23]

Ralph Chill and Gabriele Nebe. Correction to: the cremona problem in dimension 2. Archiv der Mathematik, 2023. Records withdrawal of the conclusions. URL: https://doi.org/10.1007/s00013-023-01863-0, doi:10.1007/s00013-023-01863-0.

[dBvdE05]

Michiel de Bondt and Arno van den Essen. A reduction of the jacobian conjecture to the symmetric case. Proceedings of the American Mathematical Society, 133(8):2201–2205, 2005. URL: https://doi.org/10.1090/S0002-9939-05-07570-2, doi:10.1090/S0002-9939-05-07570-2.

[Druzkowski83]

Ludwik M. Drużkowski. An effective approach to keller's jacobian conjecture. Mathematische Annalen, 264:303–313, 1983. URL: https://link.springer.com/article/10.1007/BF01459126, doi:10.1007/BF01459126.

[DP18]

Adrien Dubouloz and Karol Palka. The jacobian conjecture fails for pseudo-planes. Advances in Mathematics, 339:248–284, 2018. URL: https://arxiv.org/abs/1701.01425, doi:10.1016/j.aim.2018.09.020.

[GGHV22]

Jorge Alberto Guccione, Juan José Guccione, Rodrigo Horruitiner, and Christian Valqui. Increasing the degree of a possible counterexample to the jacobian conjecture from 100 to 108. 2022. Preprint. URL: https://arxiv.org/abs/2204.14178, arXiv:2204.14178.

[Gut62]

A. Gutwirth. The action of an algebraic torus on the affine plane. Transactions of the American Mathematical Society, 105(3):407–414, 1962. URL: https://www.ams.org/journals/tran/1962-105-03/S0002-9947-1962-0141664-0/, doi:10.1090/S0002-9947-1962-0141664-0.

[Jel93]

Zbigniew Jelonek. The set of points at which a polynomial map is not proper. Annales Polonici Mathematici, 58(3):259–266, 1993. URL: https://matwbn.icm.edu.pl/ksiazki/apm/apm58/apm5834.pdf.

[Jia26]

Andy Jiang. X post giving the projective symmetric-product marked-root formulation. X post by @davikrehalt, 2026. Posted 20 July 2026 at 12:02:19 UTC; the post explicitly presents the formulation as GPT output. URL: https://x.com/davikrehalt/status/2079175065695035442.

[Kel39]

Ott-Heinrich Keller. Ganze cremona-transformationen. Monatshefte für Mathematik und Physik, 47:299–306, 1939. URL: https://link.springer.com/article/10.1007/BF01695502, doi:10.1007/BF01695502.

[KParusinskiPuaunescu05]

T.-C. Kuo, Adam Parusiński, and Laurentiu Păunescu. A proof of the plane jacobian conjecture. 2005. Withdrawn. URL: https://arxiv.org/abs/math/0509431, arXiv:math/0509431.

[LL24]

Kyungyong Lee and Li Li. On the two-dimensional jacobian conjecture: magnus' formula revisited, iv. 2024. Preprint. URL: https://arxiv.org/abs/2408.01279, arXiv:2408.01279.

[Lit26a]

Daniel Litt. X comment identifying the projective and coordinate presentations. X comment by @littmath, 2026. Posted 21 July 2026; describes the Jiang and Alpöge posts as the same construction. URL: https://x.com/littmath/status/2079475742401003816.

[Lit26b]

Daniel Litt. X thread explaining the marked-root source through an affine-line bundle and pencils of lines. X thread by @littmath, 2026. Posted 20 July 2026; initial bundle observation at https://x.com/littmath/status/2079347618417971212 and ChatGPT-assistance disclosure at https://x.com/littmath/status/2079414119720165633. URL: https://x.com/littmath/status/2079353531430289734.

[Lou26]

Aaron Lou. Deriving an explicit polynomial counterexample to the jacobian conjecture: a reproducible cubic-factor and resultant construction. 2026. Research note, 20 July 2026; author X profile @aaron_lou: https://x.com/aaron_lou. URL: https://aaronlou.com/jacobian_counterexample_derivation.pdf.

[ML21]

Leonid Makar-Limanov. On the newton polyhedron of a jacobian pair. Izvestiya: Mathematics, 85(3):457–467, 2021. URL: https://arxiv.org/abs/2106.06869, doi:10.1070/IM9067.

[ML25]

Leonid Makar-Limanov. On the shape of a counterexample to the two-dimensional jacobian conjecture. Serdica Mathematical Journal, 51:299–314, 2025. URL: https://serdica.math.bas.bg/index.php/serdica/article/download/300/153/862, doi:10.55630/serdica.2025.51.299-314.

[MLT24]

Leonid Makar-Limanov and Anna Trakhtenberg. Properties of a jacobian mate. 2024. MPIM Preprint 2024-33. URL: https://archive.mpim-bonn.mpg.de/id/eprint/5148/1/mpim-preprint_2024-33.pdf.

[Miy05]

Masayoshi Miyanishi. Affine pseudo-coverings of algebraic surfaces. Journal of Algebra, 294(1):156–176, 2005. URL: https://www.sciencedirect.com/science/article/pii/S0021869305000657, doi:10.1016/j.jalgebra.2005.01.042.

[Miy15]

Masayoshi Miyanishi. Lectures on geometry and topology of polynomials—surrounding the jacobian conjecture. 2015. Lecture notes; originally delivered in 2003. URL: https://arxiv.org/abs/1504.07179, arXiv:1504.07179.

[Mos24]

Vered Moskowicz. There are no keller maps having prime degree field extensions. 2024. Preprint; treated provisionally in this audit. URL: https://arxiv.org/abs/2407.13795, arXiv:2407.13795.

[Ore87]

Sergei Yu. Orevkov. On three-sheeted polynomial mappings of ℂ². Mathematics of the USSR-Izvestiya, 29(3):587–596, 1987. URL: https://www.math.univ-toulouse.fr/~orevkov/jc86.pdf, doi:10.1070/IM1987v029n03ABEH000984.

[RodriguezDiaz26]

Lázaro Orlando Rodríguez Díaz. On the origin of the jacobian conjecture. Comptes Rendus Mathématique, 364:363–370, 2026. URL: https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.831/, doi:10.5802/crmath.831.

[Tru26]

Tuyen Trung Truong. On the properness of polynomial selfmaps. Vietnam Journal of Mathematics, 2026. URL: https://link.springer.com/article/10.1007/s10013-026-00804-y, doi:10.1007/s10013-026-00804-y.

[Tsu05]

Yoshifumi Tsuchimoto. Endomorphisms of weyl algebra and p-curvatures. Osaka Journal of Mathematics, 42(2):435–452, 2005. URL: https://ir.library.osaka-u.ac.jp/repo/ouka/all/7472/, doi:10.18910/7472.

[VGG17]

Christian Valqui, Jorge A. Guccione, and Juan J. Guccione. On the shape of possible counterexamples to the jacobian conjecture. Journal of Algebra, 471:13–74, 2017. URL: https://arxiv.org/abs/1401.1784, doi:10.1016/j.jalgebra.2016.08.039.

[Var99]

Yakov Varshavsky. Infinite-dimensional algebraic varieties and proof of the jacobian conjecture. 1999. Withdrawn. URL: https://arxiv.org/abs/math/9912196, arXiv:math/9912196.

[Wan80]

Stuart Sui-Sheng Wang. A jacobian criterion for separability. Journal of Algebra, 65(2):453–494, 1980. URL: https://www.sciencedirect.com/science/article/pii/0021869380902331, doi:10.1016/0021-8693(80)90233-1.

[Wri93]

David Wright. The jacobian conjecture: linear triangularization for cubics in dimension three. Linear and Multilinear Algebra, 34(2):85–97, 1993. URL: https://www.tandfonline.com/doi/abs/10.1080/03081089308818214, doi:10.1080/03081089308818214.

[Xu22]

Quan Xu. A proof of the generalized jacobian conjecture. 2022. Unrefereed claimed proof; not accepted as a resolution. URL: https://arxiv.org/abs/2209.01451, arXiv:2209.01451.

[Zolkadek08]

Henryk Żołądek. An application of newton–puiseux charts to the jacobian problem. Topology, 47(6):431–469, 2008. URL: https://doi.org/10.1016/j.top.2008.04.001, doi:10.1016/j.top.2008.04.001.