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Varieties decompose uniquely into finitely many irreducibles, and each variety is generated by only finitely polynomials. These two finiteness properties make varieties seemingly "manageable" objects, and leads me to the question:

Can a computer, given a variety (finite set of polynomials) produce a list of its irreducible components (finite set of finite sets of polynomials) in finite time?

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This is just about primary decomposition! There are several CAS which can do that, for example Singular.

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  • $\begingroup$ Thanks, do you know if Mathematica has such a function? $\endgroup$
    – Randomblue
    Commented Oct 25, 2009 at 12:23
  • $\begingroup$ I don't know Mathematica because I hate this Sin[] notation and so on :) Maple can do primary decompositions in polynomial rings (theoretically, I never used this). I think Sage just uses the interface to Singular. $\endgroup$
    – user717
    Commented Oct 25, 2009 at 12:41
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    $\begingroup$ Magma can do it too. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 25, 2009 at 13:36
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    $\begingroup$ As can Macaulay2. $\endgroup$
    – Graham Leuschke
    Commented Oct 25, 2009 at 15:10

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