  This paper considers the problems of finite determinacy and approximation of
flat analytic maps from germs of real or complex analytic spaces. It is shown
that the flatness of analytic maps from germs of real or complex analytic
spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it
is shown that flat maps from complete intersection and Cohen-Macaulay analytic
germs can be arbitrarily closely approximated by algebraic and Nash maps
respectively in such a way that the Hilbert-Samuel function of the special
fibre is preserved. It is also proved that in the complex case the preservation
of the Hilbert-Samuel function implies the preservation of Whitney's tangent
cone.
