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.