In singularity theory it is of interest to know whether one can approximate a germ of a real or complex analytic space by germs that are algebraic or Nash, and which are equisingular.

In this talk I consider the problem of the approximation of real or complex analytic germs by germs that are Nash and that are equisingular with respect to the Hilbert-Samuel function. The motivation for the choice of the Hilbert-Samuel function is the central role that it plays in Hironaka's resolution of singularities, where it serves as a measure of singularity. In this talk, I will present two main results. The first is that a Cohen-Macaulay analytic germ can be arbitrarily closely approximated by Nash germs which are also Cohen-Macaulay and share the same Hilbert-Samuel function. The second is that every analytic germ can be arbitrarily closely approximated by a Nash germ that is topologically equivalent, and has the same Hilbert-Samuel function. The proof of these results rests upon encoding the algebro-geometric properties that are to be preserved by the approximation in a system of polynomial equations. The approximations can then be obtained by applying Artin's approximation theorem. I will present the proof of the approximation result for Cohen-Macaulay analytic germs in order to demonstrate the technique. Key ingredients of the proof are Hironaka's diagram of initial exponents and a generalisation of Buchberger's criterion to the case of standard bases of power series due to T. Becker in 1990. At the end of the talk I shall comment on future work on equiresolvable approximations for which the results presented in this talk are a key step. This talk is based on joint work with Janusz Adamus at the University of Western Ontario.