The generalization of Buchberger's algorithm presented in this paper has many
applications. We just mention the computation of Milnor numbers, Tjurina
numbers, local multiplicities, Buchsbaum-Rim and Polar multiplicities, first
and second order deformations of isolated singularities, projections of
families with affine fibres onto a local base space and, of course, all the
usual ideal theoretic operations in a factorring
*Loc*_{;}*SPMlt*;*K*[*x*]/*I* such
as intersection, ideal quotient and decision about ideal or radical
membership. For further applications see [AMR]. Here we shall only explain how its implementation in SINGULAR
helped to find a partial answer to Zariski's multiplicity question.

Let
,
,
*f*(0) = 0,
be a not constant convergent powerseries and mult
the multiplicity of *f* (for the general definition of multiplicity
see the end of this chapter). Let
be a sufficiently
small ball with centre 0 and
,
the
hypersurface singularity defined by *f*. If
is another
powerseries and
,
then *f* and *g* (or
*X* and *Y*) are called topologically equivalent if there exists a
homeomorphism
such that *h*(*X*) = *Y*. The topological
type of *f* is its class with respect to topological equivalence.

Zariski asked in 1971 (cf. [Z]) whether two complex hypersurface
singularities *f* and *g* with the same topological type have the same
multiplicity.

Zariski's question (usually called Zariski's conjecture) is, in general, unsettled but the answer is known to be yes in the following special cases:

- -
*n*= 2, that is for plane curve singularities (Zariski, Lê Dung Trang),- -
*f*is semiquasihomogeneous and*g*is a deformation of*f*(Greuel [Gr], O'Shea [OS]).

Recall that *f* is called semiquasihomogeneous if there exists an analytic change of
coordinates and positive weights such that the sum of terms of smallest
weighted degree has an isolated singularity.

The two series of examples
*f*_{t}^{a, b, c} and
*g*_{t}^{a, b, c, d, e
} in Chapter 3 were actually constructed to find a counter example to
Zariski's conjecture. The idea is as follows: let

be a deformation of

denote the Milnor number of

For the above mentioned series *f*_{t}^{abc} and
*g*_{t}^{abcde} the
multiplicity is not constant. For *t* = 0 both series have non-degenerate
Newton diagram and one can show that
and
and small and some restrictions on
*a*, *b*, *c*, *d*, *e*). Since
is
several hundred or even several thousand there seemed to be a good chance for
or
to be constant. Using
SINGULAR we were able to compute many of these Milnor numbers but neither
nor
were constant.
(Actually, in most cases we obtained
and
.) None of the
existing computer algebra systems were able to compute the standard basis of
the ideal of partials of *f*_{t} respectively *g*_{t} for relevant cases. (Only the
system Macaulay was able to do some cases with small *a*, *b*, *c* using Lazard's
method but it needed hours or days, whereas SINGULAR needed seconds or
minutes. In these cases the success of SINGULAR was mainly due to the
HCtest.) The failure to find a counter example led to the following positive
result which shows that the families *f*_{t} and *g*_{t} can never be a counter
example.

Let *f*_{t}(*x*) be a (1-parameter) holomorphic **family of isolated
hypersurface singularities**, that is
is an isolated critical
point of *f*_{t} for each *t* close to
.
The **polar curve** of
such a family is the curve singularity in
defined by the
ideal
.

**Proof**: We may assume that
and then the polar
curve(*f*_{t}|*H*) is given by
while polar curve
is given by
.
Hence, the
assumption is equivalent to
.

We shall use the valuation test for -constant by Lê and Saito
([LS]):

constant
for any holomorphic curve
we have val
,
.
Moreover, this is equivalent to ``'' replaced by
``>''. (val denotes the natural valuation with respect to *s*.)

Now let be any curve in . Then implies that val , .

Applying the valuation test to *f*_{t} and to ,
the result follows.

**Proof**: Since *f*_{t} has an isolated singularity we may add
terms of sufficiently high degree without changing the analytic type of *f*_{t}.
If *n* = 3 we may replace *g*_{t} by
*g*_{t}(*x*_{1}, *x*_{2}) + *x*_{1}^{N} + *x*_{2}^{N}, *N*
sufficiently big, which has an isolated singularity and the
same multiplicity as
*g*_{t}(*x*_{1}, *x*_{2}). Hence, in any case we may assume that *g*_{t} has an
isolated singularity. Applying the preceding lemma to the hyperplane
we obtain
constant.
But since Zariski's conjecture is true for plane curve singularities and for
deformations of semiquasihomogeneous singularities ([Gr], [OS]), mult(*g*_{t}) is constant.

The Milnor number
of an isolated singularity can be computed as the
number of monomials in
where *I* is the
leading ideal of
with respect to any ordering > such that *x*_{i} < 1,
.
This follows from the following Corollary 5.4.

The reason why standard bases can be applied to compute certain invariants of
algebraic varieties or singularities (given in terms of submodules
), is that for any monomial ordering on *K*[*x*]^{r} we have:

- a)
- (Loc
_{;}*SPMlt*;*K*[*x*])^{r}/*I*is a (flat) deformation of Loc_{;}*SPMlt*;*K*[*x*]^{r}/*L*(*I*) (as we shall show below) which implies that certain invariants behave semicontinuously or even continuously during the deformation. - b)
- For a monomial ideal these invariants can be computed combinatorially (but one needs extra algorithms for the actual computation).

In order to show a) we make the following construction:

Let
be a standard basis of
.
Any monomial
may be
identified with the point
.
For a weight vector
we define

to be the weighted degree of . Let in

It is not difficult to see that there exists a weight vector
(indeed almost all *w* will do) such that in
_{w}(*g*_{i}) = *c*(*g*_{i}) *L*(*g*_{i}),
,
and, moreover, in
_{w}(*I*) = *L*(*I*).

We choose such a *w* and shall now construct the deformation from *L*(*I*) to
*I*:

For
we can write
such
that the weighted degree of each monomial of
is .
Let *t* be
one extra variable and put

Let
be the submodule generated by all ,
.
On *K*[*x*,*t*]^{r} we choose the product ordering with lex^{-} on *K*[*t*]:
if *p* > *q* or if *p* = *q* and
.

With respect to this ordering we have and, moreover, . In particular, is a standard basis of .

Let
,
and *K*(*t*) the
quotient field of *K*[*t*].

and ``generic fibre''

**Proof**: The statements regarding the special and the generic
fibres are easy. If
then the support of
is
surjective over Spec
*K*[*t*]_{(t)} and hence ([AK], V, Proposition 2.4)
it remains to show that *t* is a non-zero divisor of
.
Let
and
.
By Corollary
1.11 we have

and if . Hence,

**Proof**: If < is a wellordering, the monomials not in *L*(*I*) are
a basis of the free module
(Theorem of Macaulay cf. [E]), hence the result.
In general, it is easy to see that these monomial are linear independent
modulo *IR*. (Use a standard basis of *I* and Corollary 1.11.)

Hence, if *R*^{r}/*IR* is finite dimensional, there are only finitely many monomials
in
.
The proposition implies that
is
*K*[*t*]_{(t)}-free with these monomials as basis, hence they also generate
*R*^{r}/*IR*.

**Proof**: *I* = *R*^{r} implies
*L*(*I*) = *K*[*x*]^{r}, hence we may assume
.
Faithful flatness implies that
([AK], V, Proposition 2.10), hence the result.

Let us finish with a final remark about multiplicites in the local case:

Consider the local ring
*R* = *K*[*x*]_{(x)} with maximal ideal
and
*M* = *R*^{r}/*IR* a finitely generated *R*-module, where *I* is
given as a submodule of *K*[*x*]^{r} by finitely many generators. Consider

which is a graded module over . The formal power series

is called the

If denotes the Hilbert function of the graded module we have (cf. [Ma]) and a polynomial (the Hilbert polynomial of ) of degree

such that for

It follows that

and there exists a polynomial of degree

where and is called the (Samuel)