Finite determinacy and approximation of flat maps

We show that the property of flatness of maps from germs of real or complex analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, we also show the existence of Nash approximations to flat maps from such germs that preserve the Hilbert-Samuel function of the special fibre.

On finite determinacy of complete intersection singularities

We give an elementary combinatorial proof of the following fact: Every real or complex analytic complete intersection germ X is equisingular -- in the sense of the Hilbert-Samuel function -- with a germ of an algebraic set defined by sufficiently long truncations of the defining equations of X.