  The automorphism groups ${\rm Aut\,}A_n$ and ${\rm Aut\,}W_n$ of the
polynomial algebra $A_n=C[x_1,x_2,\cdots, x_n]$ and the rank $n$ Witt algebra
$W_n={\rm Der\,}A_n$ 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 ${\rm End\,}W_n\setminus\{0\}$
of nonzero endomorphisms of $W_n$ via the semigroup of the so-called Jacobi
tuples, we establish an isomorphism between ${\rm Aut}\,A_n$ and ${\rm
Aut\,}W_n$ for any positive integer $n$. In particular, this enables us to work
out the automorphism group ${\rm Aut\,}W_2$ of $W_2$.
