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NoetherianOperators :: Strategy => "PunctualHilbert"

Strategy => "PunctualHilbert" -- strategy for computing Noetherian operators

This strategy implements Algorithm 3.8 in Primary ideals and their differential equations.

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
i5 : isPrimary Q

o5 = true
i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualHilbert")
 -- 0.0836308 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

See also