Automorphism groups of Witt algebras

The automorphism groups $\mathrm{A}\mathrm{u}\mathrm{t}{A}_n$ and $\mathrm{A}\mathrm{u}\mathrm{t}{W}_n$ of the polynomial algebra $A_n=C[x_1,x_2,\cdots ,x_n]$ and the rank $n$ Witt algebra $W_n=\mathrm{D}\mathrm{e}\mathrm{r}{A}_n$ are studied in this paper. It is well-known that ${\rm Aut,}A_n$for$n\ge3$and${\rm Aut,}W_n$for$n\ge2$ are open. In the present paper, by characterizing the semigroup $\mathrm{E}\mathrm{n}\mathrm{d}{W}_n\setminus \{0\}$ of nonzero endomorphisms of $W_n$ via the semigroup of the so-called Jacobi tuples, we establish an isomorphism between $\mathrm{A}\mathrm{u}\mathrm{t}{A}_n$ and ${\rm Aut,}W_n$foranypositiveinteger$n$. In particular, this enables us to work out the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}{W}_2$ of $W_2$.