The first application is a counterexample to a conjectured generalisation of a theorem of Saito which says that, for an isolated hypersurface singularity, the exactness of the Poincaré complex implies that the defining polynomial is, after some analytic coordinate change, weighted homogeneous.

**Theorem** [Saito1971]:

*
If
has an isolated singularity at 0, then the
following are equivalent:*

*is weighted homogeneous for a suitable choice of coordinates.**where is the Milnor number and*

the Tjurina number.*The holomorphic Poincaré complex**is exact.*

A natural problem is whether the theorem holds also for complete intersections with . Again we have a Milnor number and a Tjurina number ,

** Theoretical reduction** [Greuel, Martin and Pfister1985]:

*If is a complete intersection of dimension 1, then (1)
(2)
(3).*

*If , then (3)
(2) if
and if
are weighted homogeneous.*

PS showed that (3)
(2) does ** not** hold in general:

The proof uses an implementation of the standard basis algorithm in a forerunner of SINGULAR and goes as follows:

- Compute to show that is exact;
- compute .

One obtains , that is, is not weighted homogeneous.

To do this we must be able to compute standard bases of modules over local rings.

The counterexample was found through a computer search in a list of singularities classified by Wa.