On polynomial automorphisms of Nagata type

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 {\L}ojasiewicz exponent at infinity.