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Given a system of polynomial equations in several complex variables, is there a way to determine if the solution space has disconnected components using some computational algebra system like Mathematica, Magma, Macauly2, Singular, etc?

In case it is helpful: The system of equations I am interested in is small enough that I can calculate the Groebner basis over the rationals, the solution space is not zero-dimensional, and (after a long compute) Singular was able to give the prime decomposition of the ideal generated by the system of polynomial equations.

My current plan is to now check if the parts of the decomposition share share solutions by checking if the basis for the union of their generators does not collapse to 1. But I don't know how to check if an individual part has disconnected components or not. Maybe I am over complicating this, and there is an easier method to check connectedness? Is it sufficient to check if each polynomial in the basis over the rationals is irreducible over the complex numbers as well?

PPenguin
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  • Are you looking at the projective solution space? In this case, you just need to check that the defining equations form a regular sequence, and then the solution set is connected by the lemma of Enriques-Zariski. Determining that a given sequence is regular comes down to finding the corresponding Koszul complex, which at least Macaulay2 can do. – Daniel Jan 06 '24 at 04:52
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    Every irreducible component of a complex affine variety is connected ( a theorem ). So now you have to see if the intersection graph given by the connected components is connected. – orangeskid Jan 06 '24 at 18:13
  • Related: https://math.stackexchange.com/questions/2195661/prime-ideal-implies-irreducible-affine-variety – PPenguin Jan 07 '24 at 07:31
  • Question/Answer for the connectedness of irreducible component: https://math.stackexchange.com/questions/2628535/connectedness-and-path-connectedness-of-irreducible-affine-algebraic-set-in-m – PPenguin Jan 07 '24 at 07:57

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