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 $f = \sum c_\alpha x^\alpha \in {\mathbb{C}}\{x_1, \dots, x_n\},\; f(0) = 0$, be a hypersurface singularity, and let mult$\,(f) := \min\{\vert\alpha\vert\; \Bigl\lvert\; c_\alpha \not= 0\Bigr.\}$ be the multiplicity.

We say that $f$ and $g$ are topological equivalent, $f \overset{\text{top}}{\sim} g$, if there is a homeomorphism

$$(B, f^{-1}(0) \cap B,0) \overset{\sim}{\longrightarrow } (B, g^{-1}(0) \cap B,0)$$

Zariski's conjecture may be stated as: $f\overset{\text{top}}{\sim} g \Rightarrow \text{mult}\,(f) = \text{mult}\,(g)$.

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 $f = f_0$:

$$f_t(x) = f(x) + tg(x,t), \quad \vert t\vert$$    small$$.$$

Then use the theoretical fact proved by Lê and Ramanujam:

$$f_0\overset{\text{top}}{\sim} f_t \Rightarrow \mu(f_0) = \mu(f_t)$$

(`` $\Leftarrow$'' holds also, except for $n = 3$, where the answer is still unknown) where $\mu(f_0)$ respectively $\mu(f_t)$ are the Milnor numbers.

We tried to construct a deformation $f_t$ of $f_0$ where the multiplicity mult$\,(f_t)$ drops but the Milnor number $\mu(f_t)$ is constant.

Our candidates $(a, b, c \in {\mathbb{N}})$ came from a heuristical investigation of the Newton diagram, one being the following series:

$$f_t = x^a + y^b + z^{3c} + x^{c+2} y^{c-1} + x^{c-1} y^{c-1} z^3 + x^{c-2} y^c (y^2 + tx)^2,\; a,b,c \in {\mathbb{N}}.$$

Obviously, the multiplicity drops. Computing $\mu$ with SINGULAR, we obtain for $(a, b, c) = (37, 27, 6)$: $\mu(f_0) = 4840$, $\mu(f_t) = 4834$, thus $f_0$ and $f_t$ 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

$$f_t = g_t(x,y) + z^2 h_t(x,y),$$   mult$$\,(g_t) <$$   mult$$\,(f_0).$$

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