Degree and gcd constraints on a plane counterexample

These results constrain the arithmetic of the two component degrees. That is a different invariant from the size bounded by Moh and from the field degree bounded by Campbell. Each result is the plane conjecture restricted to Jacobian pairs whose degree gcd lies in an excluded set, and proves it there.

• $d\ne1$. Magnus (1955), reproved by Nagata (1988).
• $d$ not prime. Nagata (1989), who also shows $d\ge9$ when neither degree divides the other.
• $d\ge16$. Heitmann (1990). His 1993 corrigendum reports a gap in the proof of one theorem and proves an analogous one under stronger hypotheses. The bound is secure independently, through Guccione–Guccione–Valqui's elementary proof.^[1]
• $d$ not twice a prime, and $d\le16$ excluded. Żołądek, with Newton–Puiseux charts (2008), and independently Guccione–Guccione–Valqui (2017) for twice a prime. The case $d=16$ rests on Żołądek alone: Guccione–Guccione–Valqui still study it at length, which suggests they did not regard it as closed.
• One degree a product of at most two primes. Appelgate and Onishi (1985) prove the pair is an automorphism in that case.

The same Żołądek paper proves the plane conjecture for every Keller map of topological degree at most 5, the wall drawn as rows 3–5 on the JC₂ chart.^[2] Newton-polygon methods from Abhyankar's two-points-at-infinity framework underlie the gcd arguments.

The conclusion that the least gcd not excluded is 18 is the atlas's combination of the published exclusions. No single source states it.

1. ↑ R. C. Heitmann, On the Jacobian conjecture, J. Pure Appl. Algebra 64 (1990) 35–72; corrigendum 90 (1993) 199–200. J. A. Guccione, J. J. Guccione, C. Valqui, On the shape of possible counterexamples to the Jacobian Conjecture, J. Algebra 471 (2017) 13–74.
2. ↑ H. Żołądek, An application of Newton–Puiseux charts to the Jacobian problem, Topology 47 (2008) 431–469, doi:10.1016/j.top.2008.04.001.