Finite determinacy and approximation of flat maps

This paper considers the problems of finite determinacy and approximation flat analytic maps from germs real (resp. complex) spaces to R m \mathbb {R}^m C mathvariant="double-struck">C {C}^m ). It is shown that flatness whose local rings are Cohen-Macaulay finitely determined. Further, it complete intersection Cohen-Macaulay) can be approximated arbitrarily well by polynomial Nash) in such a way Hilbert-Samuel function special fibre preserved. also proved complex case preservation implies Whitney’s tangent cone.