This strategy implements Algorithm 4.1 in the paper Primary decomposition of modules: a computational differential approach.
The following example deals with a rather non-trivial primary ideal to show the capabilities of this strategy.
i1 : R = QQ[x_1,x_2,x_3,x_4] o1 = R o1 : PolynomialRing |
i2 : k = 3 o2 = 3 |
i3 : J = ideal((x_1^2-x_2*x_3)^k,(x_1*x_2-x_3*x_4)^k,(x_2^2-x_1*x_4)^k)
6 4 2 2 2 3 3 3 3 2 2 2 2
o3 = ideal (x - 3x x x + 3x x x - x x , x x - 3x x x x + 3x x x x -
1 1 2 3 1 2 3 2 3 1 2 1 2 3 4 1 2 3 4
------------------------------------------------------------------------
3 3 6 4 2 2 2 3 3
x x , x - 3x x x + 3x x x - x x )
3 4 2 1 2 4 1 2 4 1 4
o3 : Ideal of R
|
i4 : Q = saturate(J,ideal(x_1*x_2*x_3*x_4))
5 3 2 3 4 2 2 2 2 2 2 2 3
o4 = ideal (x x + x x x - 4x x x x - x x + 3x x x x + x x x - x x x ,
2 3 1 2 4 1 2 3 4 1 4 1 2 3 4 2 3 4 1 3 4
------------------------------------------------------------------------
4 4 2 2 3 2 3 2 2 2 3 3 2 3
x x x + x x x - 3x x x x - x x x - x x x + 4x x x x - x x , x x x
1 2 3 1 2 4 1 2 3 4 2 3 4 1 3 4 1 2 3 4 3 4 1 2 3
------------------------------------------------------------------------
4 2 5 3 2 2 2 2 2 3 2 6 4
- x x + x x - 4x x x x + 3x x x x + x x x - x x x , x - 3x x x +
2 3 1 4 1 2 3 4 1 2 3 4 1 3 4 2 3 4 2 1 2 4
------------------------------------------------------------------------
2 2 2 3 3 3 3 2 2 2 2 3 3 6 4
3x x x - x x , x x - 3x x x x + 3x x x x - x x , x - 3x x x +
1 2 4 1 4 1 2 1 2 3 4 1 2 3 4 3 4 1 1 2 3
------------------------------------------------------------------------
2 2 2 3 3
3x x x - x x )
1 2 3 2 3
o4 : Ideal of R
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i5 : isPrimary Q o5 = true |
i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
-- 0.19574 seconds elapsed
o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
------------------------------------------------------------------------
2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
o6 : List
|