Inverse limits of automorphisms of truncated polynomials and applications related to Jacobian conjecture

In this note, we investigate Jacobian conjecture through investigation of automorphisms of polynomial rings in characteristic p. Making use of the technique of inverse limits, we show that under Jacobian condition for a given homomorphism φ of the polynomial ring 𝐤[x_1,…,x_n], if φ preserves the maximal ideals, then φ is an automorphism.