A variation on Magnus' theorem and its generalizations

Abstract

Let k be a field of characteristic zero, and let f : k[x, y] → k[x, y], f : (x, y) → (p, q), be a k-algebra endomorphism having an invertible Jacobian.WriteDenote the set of prime numbers by P . Under two mild conditions, we prove that, if gcd(gcd(n, deg x (an)), gcd(r, deg x (cr))) ∈ {1, 8} ∪ P ∪ 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 ) = ǫx αµ y βµ , l 1,1 (Q) = δx αν y βν , where ǫ, δ ∈ k × , 1 < α < β, d := gcd(α, β) > 1, 1 < ν < µ, gcd(µ, ν) = 1, actually satisfies d > 2.