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 $p=a_n{y}^n+\cdots +a_{1}y+a_0$, where $n=de{g}_y(p)\in \mathbb{N}$, $a_i\in k[x]$, $0\le i\le n$, $a_n\ne 0$, and $q=c_r{y}^r+\cdots +c_{1}y+c_0$, where $r=de{g}_y(q)\in \mathbb{N}$, $c_i\in k[x]$, $0\le i\le r$, $c_r\ne 0$. Denote the set of prime numbers by $P$. Under two mild conditions, we prove that, if $\gcd (\gcd (n,de{g}_x(a_n)),\gcd (r,de{g}_x(c_r)))\in \{1,8\}\cup P\cup 2P$, 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 $d>2$.

10 pages. The two versions are identical, except for the bibliography