The attempt to find a counterexample to Zariski's multiplicity conjecture -- which says that the multiplicity (lowest degree) of a power series is an invariant of the embedded topological type -- led, finally after many experiments and computations, to a partial proof of this conjecture. For this, an extremely fast standard basis computation for 0-dimensional ideals in a local ring was necessary.

The following question was posed by Za in his retiring address to the AMS in 1971.

Let , be a hypersurface singularity, and let mult be the multiplicity.

We say that and are topological equivalent, , if there is a homeomorphism

** Zariski's conjecture** may be stated as:
.

The result is known to be true for curves (Zariski, Lê) and weighted homogeneous singularities Gre3,OS.

Our attempt to find a counterexample was as follows:

Consider deformations of :

small

Then use the theoretical fact proved by Lê and Ramanujam:
We tried to construct a deformation of where the multiplicity mult drops but the Milnor number is constant.

Our candidates came from a heuristical investigation of the Newton diagram, one being the following series:

Since the Milnor numbers of possible counter examples have to be very big, we need an extremely efficient implementation of standard bases. For this, the ``highest corner'' method of GP1 was essential.

Trying many other classes of examples, we did not succeed in finding a counter example. However, an analysis of the examples led to the following

** Partial proof of Zariski's conjecture** [Greuel and Pfister1996]:

*Zariski's conjecture is true for deformations of the form*

mult mult

There is also an invariant characterisation of the deformations of the above
kind. The general conjecture is, up to today, still open.