Smale 16 — the plane Jacobian conjecture

• Slug: smale-jacobian-n2

• Status: open — dent: an exact replay excludes the published exceptional degree pair $(72,108)$, giving

  $$\max\{\deg P,\deg Q\}\ge 125$$

  for every hypothetical plane counterexample. This does not prove the conjecture.

• Area: affine algebraic geometry; polynomial automorphisms; computer algebra

• Started: 2026-08-27

In general

Let $F=(P,Q):\mathbb C^2\to\mathbb C^2$ be a polynomial map. The plane Jacobian conjecture says that if

then $F$ is invertible and its inverse is polynomial. Linear changes and a rescaling reduce the nonzero constant to 1.

The conjecture remains open in two variables. The 2026 announcements of higher-dimensional counterexamples are not treated here as settled, and in any case do not settle the plane problem.

Published finite record

Moh excluded plane counterexamples through degree 100. Guccione, Guccione, Horruitiner and Valqui subsequently proved that a counterexample must satisfy one of

Their Proposition 4.3 turns the exceptional pair into two explicit systems of Laurent-polynomial coefficient equations with bracket $[P,Q]=x^2$. A 2026 archive by Billel Helali gives exact characteristic-zero certificates that both systems generate the unit ideal. The q1 verifier pins that archive, replays all of its identities, and independently checks the bridge from the published Newton polygons to the coefficient systems.

Precise finite question attacked here

Are either of the two Laurent systems in Proposition 4.3 consistent over an algebraically closed field of characteristic zero? If both are inconsistent, the exceptional degree pair disappears and the next possible maximum degree is at least 125.

What would count as a new bound

The record before this attack still allowed maximum degree 108. A dent is an exact, reproducible exclusion of both $(72,108)$ systems, together with a check of the reduction and normalization that connect them to the published degree list. The resulting inequality must be written as

An unfinished Gröbner-basis calculation, a modular inconsistency, or a search over only part of either coefficient system is residue and changes no bound.

Sources

• Keller, Ganze Cremona-Transformationen (1939)
• Bass–Connell–Wright, degree reduction and formal inverse (1982)
• Moh, On the Jacobian conjecture and the configurations of roots (1983)
• Guccione–Guccione–Horruitiner–Valqui, arXiv:2204.14178v1
• Helali, exact $(72,108)$ certificate archive