Jet groups

Let N⩾2. We describe some classes of subgroups of the n-jet group of local analytic diffeomorphisms of (CN,0). We apply the results to show that if G is any group of algebraic automorphisms strictly containing the affine group, then any algebraic automorphism can be approximated at the origin and at any order by an element of G. We also show that any algebraic or analytic automorphism can be interpolated at any order and at any finite set of points, by a tame one.