Why are points counted as a single component in numerical_irreducible_decomposition
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#586
Replies: 2 comments 1 reply
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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 I agree, we could add a warning in the documentation. |
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Instead of just printing 4, maybe add some more text like: |
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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:
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?
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