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I only know how to find the irreducible components when I know what the image is, but there are lots of equations that are hard to figure out their images, is there any systematic way to find the irreducible components?

Damon
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  • Hint: think of these equations as being in $\mathbb{R}^3$ and try graphing the zero set of each equation. – rfauffar Nov 20 '14 at 14:32
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    So in the given example you were able to work out the irreducible components (perhaps by "elimination" by hand), but you want to know if there's a systematic way to do this? – hardmath Nov 20 '14 at 14:32
  • I mean in this special case, I can graph the zero set of each one easily, how about some case that equations are too difficult to graph, even if it is still hard by eliminating – Damon Nov 20 '14 at 15:09
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    Graphs won't help here, especially when you only look at solutions in $\mathbb{R}$. – Martin Brandenburg Nov 20 '14 at 15:39

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The general question "how to find irreducible components?" has a long answer. Here is an extract of the book "Ideals, varieties and algorithms" by Cox, Little, O'Shea, which points you to the literature where algorithms are presented.

It would be nice to have more general methods that could be applied to any ideal. Theorems 2,4,5, and 6 tell us that certain decompositions exist, but the proofs give no indication how to find them. The problem is that the proofs rely on the Hilbert Basis Theorem, which is intrinsically nonconstructive. Based on what we have seen in §§5 and 6, the following questions arise naturally:

$\bullet$ (Primality) Is there an algorithm for deciding if a given ideal is prime?

$\bullet$ (Irreducibility) Is there an algorithm for deciding if a given affine variety is irreducible?

$\bullet$ (Decomposition) Is there an algorithm for finding the minimal decomposition of a given variety or radical ideal?

The answer to all three questions is yes, and descriptions of the algorithms can be found in the works of HERMANN (1926), MINES, RICHMAN, and RUITENBERG (1988), and SEIDENBERG (1974, 1984). As in §2, the algorithms in these articles are not very practical. However, the work of GIANNI, TRAGER, and ZACHARIAS (1988) has recently led to algorithms implemented in AXIOM and REDUCE that answer the above questions. See also Chapter 8 of BECKER and WEISPFENNING (1993) and, for the primality algorithm, §4.4 of ADAMS and LOUSTAUNAU (1994). A different algorithm for studying these questions, based on ideas of EISENBUD, HUNEKE and VASCONCELOS (1992), has been partially implemented in Macaulay.

In the example, however, we may use the following calculation:

$(x^2+y^2+z^2,x^2-y^2-z^2+1)\\=(x^2+y^2+z^2,(-y^2-z^2)-y^2-z^2+1)\\=(x^2+y^2+z^2,y^2+z^2 - 1/2)\\=(x^2 + 1/2,y^2+z^2 - 1/2)\\=(x^2-c,y^2+z^2+c)$

where $c = -1/2$. What follows, works for any $c \in k^*$. Assuming that $k$ is algebraically closed, we factor $x^2-c=(x+\sqrt{c})(x-\sqrt{c})$ and the ideals $(x+\sqrt{c}),(x-\sqrt{c})$ are coprime. Hence, the ideal becomes $(x+\sqrt{c},y^2+z^2+c) \cap (x-\sqrt{c},y^2+z^2+c)$ and both these ideals are prime, because $k[x,y,z]$ modulo them is (in each case) $k[y,z]/(y^2+z^2+c)$, which is an integral domain since $y^2+z^2+c$ is irreducible (apply Eisenstein's criterion with $z + \sqrt{-c}$).

Community
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Martin Brandenburg
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  • I am sorry, maybe I am wrong, but it seems to me that you are using something like $I+(J\cap K)=(I+J)\cap (I+k)$, so a distributivity law for the sum, when $J,K$ are coprime. Which are the hypothesis to use this result? – Lios Mar 05 '19 at 13:50
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    @Yecabel When $J,K$ are coprime, then $I + (J \cap K) = (I + J) \cap (I + K)$ indeed holds. You can derive this from the isomorphism $R/(J \cap K) \cong R/J \times R/K$ by taking the image of the ideal $(I + J \cap K)/(J \cap K)$ and then the preimage again. – Martin Brandenburg Dec 22 '19 at 14:45