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## 2. Zariski's multiplicity conjecture

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:

( '' holds also, except for , where the answer is still unknown) where respectively are the Milnor numbers.

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:

Obviously, the multiplicity drops. Computing with SINGULAR, we obtain for : , , thus and are (unfortunately) not topologically equivalent.

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.

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