2018/10/18 by Vered Moskowicz, Moskowicz, Vered
Mathematics · #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1810.08202
openalex publication_date 2018/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. Write p=anyn+⋯+a1y+a0, where n=degy(p) ∈ ℕ, ai ∈ k[x], 0 ≤ i ≤ n, an ≠ 0, and q=cryr+⋯+c1y+c0, where r=degy(q) ∈ ℕ, ci ∈ k[x], 0 ≤ i ≤ r, cr ≠ 0. Denote the set of prime numbers by P. Under two mild conditions, we prove that, if gcd(gcd(n,degx(an)),gcd(r,degx(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, l1,1(P)=εxαμyβμ, l1,1(Q)=δxανyβν, where ε,δ∈ k×, 1 < α 1, 1 < ν< μ, gcd(μ,ν)=1, actually satisfies d > 2.