  Let $D$ be a bounded domain in $C^n$. By the theorem of H.~Cartan, the group
$Aut(D)$ of all biholomorphic automorphisms of $D$ has a unique structure of a
real Lie group such that the action $Aut(D)\times D\to D$ is real analytic.
This structure is defined by the embedding $C_v\colon Aut(D)\hookrightarrow
D\times Gl_n(C)$, $f\mapsto (f(v), f_{*v})$, where $v\in D$ is arbitrary. Here
we restrict our attention to the class of domains $D$ defined by finitely many
polynomial inequalities. The appropriate category for studying automorphism of
such domains is the Nash category. Therefore we consider the subgroup
$Aut_a(D)\subset Aut(D)$ of all algebraic biholomorphic automorphisms which in
many cases coincides with $Aut(D)$. Assume that $n>1$ and $D$ has a boundary
point where the Levi form is non-degenerate. Our main result is theat the group
$Aut_a(D)$ carries a unique structure of an affine Nash group such that the
action $Aut_a(D)\times D\to D$ is Nash. This structure is defined by the
embedding $C_v\colon Aut_a(D)\hookrightarrow D\times Gl_n(C)$ and is
independent of the choice of $v\in D$.
