On polynomial automorphisms of Nagata type
Jorge A. C. Huarcaya, Joe Palacios
TL;DR
This work analyzes Nagata-type polynomial automorphisms of $F[x,y,z]$ by parameterizing a family $(f,g,h)$ with $f=x-2y\varphi - z\varphi^2$, $g=y+z\varphi$, $h=z$, where $\varphi$ ranges over $F[x,y,z]$. The central technique is solving the linear PDE $-2y\varphi_x+z\varphi_y=0$, whose general solution is $\varphi=p(xz+y^2,z)$ with $p\in F[t_1,t_2]$, leading to explicit automorphism criteria and inverses when the Jacobian determinant is a nonzero constant. The authors prove that det$J(f,g,h)$ is a nonzero constant iff $(f,g,h)$ is an automorphism and provide an explicit inverse in terms of $p$, thereby verifying the Jacobian conjecture for this Nagata-type family; they also identify conditions under which the automorphisms are wild and discuss upper semicontinuity properties of the Łojasiewicz exponent at infinity. Collectively, the results supply a concrete, PDE-based framework for automorphism criteria in three variables and suggest a new angle on the Jacobian conjecture via Nagata-type constructions.
Abstract
We define a family of polynomial ring homomorphisms generalizing the well-known Nagata automorphism. We establish necessary and sufficient conditions under which these homomorphisms are automorphisms, and verify that they satisfy the Jacobian conjecture. Additionally, we provide a necessary condition within this family to obtain wild automorphisms, and independently derive a property related to the upper semicontinuity of the Łojasiewicz exponent at infinity.