A variation on Magnus' theorem and its generalizations

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 $d > 2$.