Why are points counted as a single component in numerical_irreducible_decomposition?

[oskarhenriksson]

Something that I've noticed leads to a slight pedagogical problem in teaching situations is that when there are isolated points in a variety $$V(F)$$, then numerical_irreducible_decomposition(F) counts them as a single component (of degree equal to the number of points), which doesn't agree with some references such as §4 here.

For instance, one could expect this to return 4 (since the irreducible components are two lines and two points), but the actual output is 3:

@var x y
F = System([(y-2)*(x-1)*(x+1),(x-1)*(y-2)*(y^2-1)])
N = numerical_irreducible_decomposition(F)
n_components(N)

Since all the isolated points are given by a single witness set, I guess having this convention is "cleaner" from a programming point of view, but is there a way to also make sense of this convention mathematically? Does it appear anywhere in the literature? And could it make sense to add a small warning about it in the documentation?

[PBrdng]

Hi @oskarhenriksson, you are right. We are counting 0-dimensional components as only one component. This goes against the usual definition, where an isolated point is an irreducible component. The reason is the following. Suppose there are 4 isolated points. Then, we don't want to show (1,1,1,1) in the terminal. Instead, we show 4, which carries the same information but with a better overview. So, here the decision was in favor of design at the cost of mathematical correctness.

I agree, we could add a warning in the documentation.

[saschatimme]

Instead of just printing 4, maybe add some more text like: 4 components of degree 1

[PBrdng]

Yes, we could do that. I will make a suggestion later this week.

[PBrdng]

One week later, I pushed a new branch where I will update NID. As a first step, I decomposed the 0-dim components into points:

julia> @var x, y, z
julia> p = (x * y - x^2) + 1 - z
julia> q = x^4 + x^2 - y - 1
julia> F = [
      p * q * (x - 3) * (x - 5)
      p * q * (y - 3) * (y - 5)
      p * (z - 3) * (z - 5)
  ]
julia> nid(F)
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) │
╰───────────┴──────────────────────────╯

I will keep working on this branch in the next weeks adding performance updates. Feel free to jump in if you have time.

[PBrdng]

The most recent version of HC.jl counts 0-dimensional components correctly.

[oskarhenriksson]

Very nice! Thanks!