Numerical Irreducible Decomposition
A numerical irreducible decomposition of a system of polynomial equations $F(x)$ is the decomposition of the zero set of $F$ into irreducible components.
Careful: The implementation is not yet final and should be considered experimental.
For example, the following code decomposes the zero set of a system that consists of a hypersurface of degree 2, two curves of degree 4 and eight points.
using HomotopyContinuation
@var x, y, z
p = (x * y - x^2) + 1 - z
q = x^4 + x^2 - y - 1
F = [p * q * (x - 3) * (x - 5);
p * q * (y - 3) * (y - 5);
p * (z - 3) * (z - 5)]
N = numerical_irreducible_decomposition(F)
The output is as follows.
Numerical irreducible decomposition with 11 components
• 1 component(s) of dimension 2.
• 2 component(s) of dimension 1.
• 8 component(s) of dimension 0.
degree table of components:
╭───────────┬──────────────────────────╮
│ dimension │ degrees of components │
├───────────┼──────────────────────────┤
│ 2 │ 2 │
│ 1 │ (4, 4) │
│ 0 │ (1, 1, 1, 1, 1, 1, 1, 1) │
╰───────────┴──────────────────────────╯
Notice that in dimension 0 the 8 points are summarized as one component (a single point is always an irreducible component).
One can also use
nid(F)
This will compute exact same thing as numerical_irreducible_decomposition(F).
See also the documentation.
Regeneration
The algorithm consists of 2 steps. The first step is u-regeneration by Duff, Leykin and Rodriguez. The output of the regeneration algorithm is a vector of witness sets, one for each dimension. The witness set $W_i$ in dimension $i$ gives $\{F=0\} \cap L_i$ where $L_i$ is a random linear space of codimension $i$; i.e., $W_i$ represents the union of irreducible components in dimension $i$.
R = regeneration(F)
3-element Vector{WitnessSet}:
Witness set for dimension 2 of degree 2
Witness set for dimension 1 of degree 8
Witness set for dimension 0 of degree 8
The second step decomposes the output of regeneration into irreducible components using monodromy.
decompose(R)
The output of decompose(R) is then the same as the output of nid(F).