Multivariate factorization internally uses random evaluation points. Therefore the way the factors are ordered is not necessarily the same for different initial random seeds.
When adding a new polynomial, the factorizing Gröbner basis algorithm
proceeds by factorizing this polynomial,say f=f1f2,
into its factors f1 and f2. Then it reenters, again,
this time for
Thus the algorithm might return a decomposition into a larger or smaller
number of components, depending on the random seed.
V(I,f1)
and
V(I,f2) \ V(f1).
General remarks on facstd:
facstd
has the same radical as the original ideal.
facstd is usually not
a complete decomposition into the components of the variety,
it only cuts the decomposition problem into smaller and hence easier
problems.
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