The plane problem after the threefold example

The plane problem after the threefold example#

Current verdict

The two-dimensional Jacobian conjecture is not solved here. As of 2026-07-20, no accepted proof or counterexample in dimension two was found in the literature audit. This chapter separates established restrictions, preprint claims, and the new exact calculations in this repository.

The threefold map has geometric degree three. That mechanism does not simply drop a variable: in the plane, geometric degrees \(2,3,4,5\) are excluded in the established literature, so the first possible degree of a noninvertible Keller map is

\[ \mu=[\mathbb C(x,y):\mathbb C(P,Q)]\geq 6. \]

The low-sheeted conclusion is cited through Żołądek’s Theorem 6.12 and is used as established input by Makar-Limanov’s refereed 2021 paper [Makar-Limanov, 2021, Orevkov, 1987, Żołądek, 2008]. Żołądek’s separate gcd-of-degrees argument has a known gap and is not used here.

A Galois function-field extension cannot be a counterexample: the Keller condition plus Galoisness forces polynomial invertibility [Campbell, 1973]. Thus a first counterexample should be sought as a composite, non-Galois cover, with degree six the first plausible target.

The factorization route in this repository produces exactly such an intermediate-cover framework. Its degree-two top extension is the standard Galois affine pseudo-covering

\[ \mathbb A^2\longrightarrow S(2,2,1)=\{u(1+uv)=w^2\}. \]

A second etale map from this pseudo-plane to \(\mathbb A^2\) would compose to a non-Galois plane Keller map of even geometric degree. Since total degrees two through five are excluded, the second map would have generic degree at least three. Miyanishi’s canonical-divisor obstruction eliminates broad classes of pseudo-planes, but vanishes for this exceptional type \(r=2\) [Miyanishi, 2005, Miyanishi, 2015]. The exceptional direction remains tied to the classical plane conjecture in the 2026 Borisov–Gabber–Vasiu manuscript [Borisov et al., 2026].

Degree and Newton constraints#

The strongest quantitative degree statement located is the 2022 preprint of Guccione–Guccione–Horruitiner–Valqui. It reports that either

\[ \max(\deg P,\deg Q)\geq 125 \]

or, up to exchanging the coordinates, the single pair

\[ (\deg P,\deg Q)=(72,108) \]

survives below \(125\) [Guccione et al., 2022]. In particular, its safe headline is a lower bound of \(108\), but it remains a preprint and is labeled that way throughout this vault.

For the exceptional pair, divide by the gcd \(36\) to obtain the primitive ratio \(2:3\). Chau’s theorem on the nonproper value set then forces all its branches to share one point at infinity and gives a cusp-type leading resultant of the form [Chau, 2004]

\[ C\left(A^3u^3-B^2v^2\right)^M \quad+ \text{terms of lower }(2,3)\text{-weight}. \]

This is why the degree-six cusp cover in Plane near-models is not an arbitrary toy: it matches the first allowed covering degree and the forced \(2:3\) geometry at infinity.

Further refereed restrictions include homothetic Newton polygons for a normalized reduced counterexample, restrictions on admissible edges, and Makar-Limanov’s field-degree inequality, which in particular gives a gap \(n-m>6\) in his notation. The 2024 integral Newton-chain work and Lee–Li’s inner-polynomial program are promising but remain preprints [Lee and Li, 2024, Makar-Limanov, 2021, Makar-Limanov, 2025, Makar-Limanov and Trakhtenberg, 2024].

Claims deliberately kept provisional#

Moskowicz’s 2024 preprint claims that prime geometric degree is impossible [Moskowicz, 2024]. It is useful motivation, but the audit found no journal publication, so no argument in this book depends on it. The stronger, established \(\mu\geq6\) restriction is enough to rule out transplanting the degree-three cover.

The 2022 paper claiming a complete proof in dimension two was formally corrected: its key proposition is false and its conclusions were withdrawn [Bartenwerfer, 2022, Chill and Nebe, 2023]. Other recent complete-proof preprints remain unaccepted. The recurring gap is control at infinity, exactly where a nonproper Keller map can lose sheets.

Working specification for a counterexample#

A credible plane counterexample must simultaneously have:

  • geometric degree at least six and non-Galois monodromy;

  • no affine ramification, although several sheets may disappear at infinity;

  • a finite normalization that contains the source \(\mathbb A^2\) as an open subset but is not itself \(\mathbb A^2\);

  • nontrivial boundary divisor classes, so deleting the ramified boundary does not create a nonconstant unit on the source;

  • the known Newton-polygon, degree, and nonproper-curve constraints.

The last three requirements are the main structural difference between a true Keller cover and the explicit near-models obtained so far.