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- W2898070830 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 $epsilon,delta in k^{times}$, $1 1$, $1 2$." @default.
- W2898070830 created "2018-10-26" @default.
- W2898070830 creator A5064387150 @default.
- W2898070830 date "2018-10-18" @default.
- W2898070830 modified "2023-09-27" @default.
- W2898070830 title "A variation on Magnus' theorem and its generalizations" @default.
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