Automorphisms of the $k$-algebra $k[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m]$

For a field $k$ of characteristic $0$, we present an algorithm for deciding if a morphism $\varphi :k[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m]\to k[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m]$ has an inverse. The algorithm also shows how to find the inverse when it exists.