Research log — Smale 16 / plane Jacobian conjecture

Papers, exact source checks, and failed lookups. Forum posts were used only to find primary artifacts; no forum number is treated as a citation.

2026-08-27 — classical record

• Keller, Ganze Cremona-Transformationen, Monatshefte für Mathematik und Physik 47 (1939), 299–306. The Springer landing page opened, but the article scan was paywalled. The bibliographic record was cross-checked against the University of Halle Keller bibliography. This is the conventional 1939 source of the conjecture.

• Bass, Connell and Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. AMS 7 (1982), 287–330. The AMS/CiNii record and PDF link and an author-uploaded full text were opened. The relevant reduction replaces a general Keller map, after stabilization, by $X-H$ with $H$ cubic homogeneous and $JH$ nilpotent; the paper also gives the rooted-tree expansion of the formal inverse.

• Drużkowski, An effective approach to Keller's Jacobian conjecture, Math. Ann. 264 (1983), 303–313. The original scan was fetched from the Göttinger Digitalisierungszentrum. Drużkowski further reduces the general problem to cubic-linear maps $X+(AX)^{*3}$. This raises the dimension and is not a low-degree classification in the original two variables.

• Moh, On the Jacobian conjecture and the configurations of roots, J. Reine Angew. Math. 340 (1983), 140–212. The publisher metadata opened, but the scan did not. Moh's author page explicitly records the computer-assisted exclusion through degree 100 and the approximate-root/Diophantine-tree data. Later papers note that the full case analysis is difficult to replay from the printed article, so this notebook uses the later explicit degree enumeration as its immediate input.

• Appelgate and Onishi, The Jacobian conjecture in two variables, J. Pure Appl. Algebra 37 (1985), 215–227. The publisher endpoint rejected an automated PDF request; the bibliographic summary records their result that the conjecture holds if either coordinate degree is a product of at most two primes. The follow-up Nowicki–Nakai paper was opened as a check on the cited lemmas.

2026-08-27 — finite degree frontier

• Nguyen, Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104, updated 27 March 2025 and published in Quaestiones Mathematicae 48(2). The HTML and abstract were replayed. It claims the complex plane conjecture through degree 104. This is compatible with, but weaker than, the more explicit degree-pair result below.

• Guccione, Guccione, Horruitiner and Valqui, Increasing the degree of a possible counterexample to the Jacobian Conjecture from 100 to 108, 29 April 2022. The abstract, Theorem 2.1, and Proposition 4.3 were checked in the arXiv HTML. Theorem 2.1 says that a counterexample has maximum degree at least 125 or degree pair $(72,108)$, up to transposition. Proposition 4.3 reduces that exception to exactly two Newton-polygon systems with $[P,Q]=x^2$.

• Billel Helali, Exact certificate replay for the JC2 degree pair (72,108), archived at Zenodo DOI 10.5281/zenodo.21479814. The Git repository was checked out at commit c530fe44e5f53b17840110931803e7c7c5a24cde. Its 86 MB replay bundle has SHA-256 232204bdb598cc2ea0368e154c8573e18bbfdc69fa631c8878de4b884b38bb18. A direct Zenodo page request failed, so the DOI, metadata, archive hash, and license were checked through the pinned Git source and its release bundle.

• Clean exact replay in a fresh Python 3.14 virtual environment passed every serialized unit-ideal certificate, the independent gmpy2 arithmetic path, the branch involution, and both hard identities. The terminal markers were ALL_SERIALIZED_EXACT_CERTIFICATES_PASS, GMPY2_EXACT_PASS, BRANCH1_EXACT_IDENTITY_PASS, BRANCH2_EXACT_IDENTITY_PASS, and JC2_72_108_EXACT_REPLAY_PASS. The independent arithmetic path evaluated 13,410 number-field products, or 335,250 scalar products.

• The transcription was checked directly against Proposition 4.3. Its Case 1 polygons contain 61 and 125 lattice points; Case 2 contains 25 and 47. With $t=xy^2$ and $z=y^{-1}$, the identities $[t,z]=-1$ and $x^2=t^2z^4$ reproduce the archive's five coefficient equations. The three chosen nonzero vertex coefficients can be normalized because the corresponding exponent matrix is nonsingular over an algebraically closed field. These checks are reimplemented in q1 in both Python and Rust.

2026-08-27 — recent approaches and claims

• Lee and Li, On the two-dimensional Jacobian conjecture: Magnus' formula revisited, IV, 2 August 2024. The paper bounds the northeastern vertex of inner Newton polygons and proves special cases of several proposed reductions. It does not give a general inverse or a stronger finite degree bound.

• Zwart, Mathieu's approach to the Jacobian Conjecture, 21 November 2025. This is an expository account of a Lie-theoretic conjecture implying the Jacobian conjecture. The implication is not a proof of the plane case.

• Rodríguez Díaz, On the origin of the Jacobian conjecture, 29 December 2025. The paper traces the statement to Kraus (1884), identifies the last step of Kraus's proof as flawed, and frames ramification at infinity as the unresolved obstruction.

• Meng and Yang, A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures, 27 July 2026, and Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, 31 July 2026, build on Alpöge's announced three-variable example. These are recent preprint claims, not community consensus in this notebook. Both say explicitly that the two-variable conjecture remains open. Gao's tangent-line sweep motivated q3, which checks why the raw sweep cannot itself be Keller in two dimensions.