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I'm trying to compute the irreducible components of an affine variety $V(U) = V(U_1) \cup V(U_2)$ . I already know that $V(U)$ is irreducible component if and only if the ideal $I(V(U))$ is prime, so, I suppose that I need to decompose the ideal into primes, but I don't know if there exists an algorithm (or an easier way to decompose varieties). Thanks in advance

Rodrigo Torres
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  • For your info, you're asking about affine varieties. I'm not 100% sure what you're asking for, but I think you're asking if there's an algorithm for decomposing an affine variety into irreducible components. If so, then the answer is probably yes, depending on how general you want to get. The answer is yes, for example for affine varieties over a field. I think you should learn some more algebraic geometry in the meantime, but you should look in to computational commutative algebra, Groebner bases in particular. – Callus - Reinstate Monica Mar 10 '18 at 03:57
  • Thanks! Yes, but the question is not about the definitions, is rather about explicit algorithms to decompose ideals on polynomial rings into prime ideals, to find the irreducible components. – Rodrigo Torres Mar 10 '18 at 04:01
  • After passing to the algebraic side (looking at the coordinate ring, or associated spec) the process of primary decomposition or localization are the general tools but I don’t know – Prince M Mar 10 '18 at 04:07
  • I have read something about primary decomposition in Atiyah-McDonald, but there isn't an explicit algorithm :( – Rodrigo Torres Mar 10 '18 at 04:13
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    Maybe this will help: https://math.stackexchange.com/a/1030933/520527. – danneks Mar 10 '18 at 07:31
  • Well! Thanks :D – Rodrigo Torres Mar 10 '18 at 14:43

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