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 :
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