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 $\phi$ of the polynomial ring $\mathbf{k}[x_1,\dots ,x_n]$, if $\phi$ preserves the maximal ideals, then $\phi$ is an automorphism.