A variation on Magnus' theorem and its generalizations

TL;DR

This paper extends Magnus' theorem by establishing new conditions under which polynomial endomorphisms with invertible Jacobian are automorphisms, and analyzes the structure of potential counterexamples to the Jacobian Conjecture.

Contribution

It introduces generalized gcd conditions that guarantee automorphisms and provides insights into the structure of known Jacobian Conjecture counterexamples.

Findings

Proves automorphism conditions based on gcd properties of polynomial degrees.

Shows that certain known counterexamples must have a gcd greater than 2.

Extends Magnus' theorem to broader classes of polynomial maps.

Abstract

Let $k$ be a field of characteristic zero, and let $f: k[x,y] \to k[x,y]$, $f: (x,y) \mapsto (p,q)$, be a $k$-algebra endomorphism having an invertible Jacobian. Write $p=a_ny^n+\cdots+a_1y+a_0$, where $n=deg_y(p) \in \mathbb{N}$, $a_i \in k[x]$, $0 \leq i \leq n$, $a_n \neq 0$, and $q=c_ry^r+\cdots+c_1y+c_0$, where $r=deg_y(q) \in \mathbb{N}$, $c_i \in k[x]$, $0 \leq i \leq r$, $c_r \neq 0$. Denote the set of prime numbers by $P$. Under two mild conditions, we prove that, if $\gcd(\gcd(n,deg_x(a_n)),\gcd(r,deg_x(c_r))) \in \{1,8\} \cup P \cup 2P$, then $f$ is an automorphism of $k[x,y]$. Removing (at least one of) the two mild conditions, we present two additional results. One of the additional results implies that the known form of a counterexample $(P,Q)$ to the two-dimensional Jacobian Conjecture, $l_{1,1}(P)=\epsilon x^{\alpha \mu}y^{\beta \mu}$, $l_{1,1}(Q)=\delta x^{\alpha \nu}y^{\beta \nu}$, where $\epsilon,\delta \in k^{\times}$, $1 < \alpha <\beta$, $d:=\gcd(\alpha,\beta) > 1$, $1 < \nu < \mu$, $\gcd(\mu,\nu)=1$, actually satisfies $d > 2$.