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NoetherianOperators :: joinIdeals

joinIdeals -- Computes the join of two ideals

Synopsis

Description

This method computes the join of two ideals. The join can be used to describe very interesting types of primary ideals that include the symbolic powers of prime ideals. For more details reader is referred to Section 7 of the paper Primary ideals and their differential equations.

i1 : R = QQ[x_1..x_9]

o1 = R

o1 : PolynomialRing
i2 : MM = genericMatrix(R, 3, 3)

o2 = | x_1 x_4 x_7 |
     | x_2 x_5 x_8 |
     | x_3 x_6 x_9 |

             3       3
o2 : Matrix R  <--- R
i3 : P = minors(2, MM)

o3 = ideal (- x x  + x x , - x x  + x x , - x x  + x x , - x x  + x x , -
               2 4    1 5     3 4    1 6     3 5    2 6     2 7    1 8   
     ------------------------------------------------------------------------
     x x  + x x , - x x  + x x , - x x  + x x , - x x  + x x , - x x  + x x )
      3 7    1 9     3 8    2 9     5 7    4 8     6 7    4 9     6 8    5 9

o3 : Ideal of R
i4 : M = ideal(x_1^2, x_2^2, x_3^2, x_4, x_5, x_6, x_7, x_8, x_9)

             2   2   2
o4 = ideal (x , x , x , x , x , x , x , x , x )
             1   2   3   4   5   6   7   8   9

o4 : Ideal of R
i5 : Q = joinIdeals(P, M)

                                                    3 3       2 2    
o5 = ideal (x x  - x x , x x  - x x , x x  - x x , x x  - 3x x x x  +
             6 8    5 9   6 7    4 9   5 7    4 8   3 8     2 3 8 9  
     ------------------------------------------------------------------------
       2     2    3 3   3   2       2           2     2    3   2     2 2    
     3x x x x  - x x , x x x  - 3x x x x x  + 3x x x x  - x x x , x x x x  -
       2 3 8 9    2 9   3 5 8     2 3 5 8 9     2 3 5 9    2 6 9   2 3 7 8  
     ------------------------------------------------------------------------
        2   2    2   2      2   2        2   2    2     2     2        
     x x x x  - x x x x  + x x x x  + x x x x  - x x x x , x x x x x  -
      1 3 7 8    2 3 7 9    1 3 8 9    1 2 7 9    1 2 8 9   2 3 4 7 8  
     ------------------------------------------------------------------------
        2   2    2            2              2   2    2     2   3 2    
     x x x x  - x x x x x  + x x x x x  + x x x x  - x x x x , x x x  -
      1 3 4 8    2 3 4 7 9    1 3 5 8 9    1 2 4 9    1 2 5 9   3 5 8  
     ------------------------------------------------------------------------
         2 2       2            3 2       2 2        2          2   2    
     3x x x x  + 3x x x x x  - x x x , x x x x  - x x x x x  - x x x x  +
       2 3 5 9     2 3 5 6 9    2 6 9   2 3 4 8    1 3 4 5 8    2 3 4 9  
     ------------------------------------------------------------------------
      2   2        2          2           3 3       2 2       2     2    3 3 
     x x x x  + x x x x x  - x x x x x , x x  - 3x x x x  + 3x x x x  - x x ,
      1 3 5 9    1 2 4 6 9    1 2 5 6 9   3 7     1 3 7 9     1 3 7 9    1 9 
     ------------------------------------------------------------------------
      3 3       2 2       2     2    3 3   3   2       2           2     2  
     x x  - 3x x x x  + 3x x x x  - x x , x x x  - 3x x x x x  + 3x x x x  -
      2 7     1 2 7 8     1 2 7 8    1 8   3 4 7     1 3 4 7 9     1 3 4 9  
     ------------------------------------------------------------------------
      3   2   3   2       2           2     2    3   2   3 2         2 2    
     x x x , x x x  - 3x x x x x  + 3x x x x  - x x x , x x x  - 3x x x x  +
      1 6 9   2 4 7     1 2 4 7 8     1 2 4 8    1 5 8   3 4 7     1 3 4 9  
     ------------------------------------------------------------------------
       2            3 2     3 2         2 2       2            3 2     3 3  
     3x x x x x  - x x x , x x x  - 3x x x x  + 3x x x x x  - x x x , x x  -
       1 3 4 6 9    1 6 9   2 4 7     1 2 4 8     1 2 4 5 8    1 5 8   3 5  
     ------------------------------------------------------------------------
         2 2       2     2    3 3     2 2        2   2    2   2      2   2  
     3x x x x  + 3x x x x  - x x , x x x x  - x x x x  - x x x x  + x x x x 
       2 3 5 6     2 3 5 6    2 6   2 3 4 5    1 3 4 5    2 3 4 6    1 3 5 6
     ------------------------------------------------------------------------
          2   2    2     2   3 3       2 2       2     2    3 3   3 3  
     + x x x x  - x x x x , x x  - 3x x x x  + 3x x x x  - x x , x x  -
        1 2 4 6    1 2 5 6   3 4     1 3 4 6     1 3 4 6    1 6   2 4  
     ------------------------------------------------------------------------
         2 2       2     2    3 3       2   3    2 2 4    3   2      
     3x x x x  + 3x x x x  - x x , x x x x x  - x x x  + x x x x x  -
       1 2 4 5     1 2 4 5    1 5   1 2 3 7 8    1 3 8    2 3 7 8 9  
     ------------------------------------------------------------------------
         2     2       2     3      4 2 2       3     2     2 2 2 2 
     4x x x x x x  + 3x x x x x  - x x x  + 3x x x x x  - 2x x x x ,
       1 2 3 7 8 9     1 2 3 8 9    2 7 9     1 2 7 8 9     1 2 8 9 
     ------------------------------------------------------------------------
          2   3    2 2   3    3                 2     2       2       2    
     x x x x x  - x x x x  + x x x x x x  - 4x x x x x x  + 3x x x x x x  -
      1 2 3 4 8    1 3 5 8    2 3 4 7 8 9     1 2 3 4 8 9     1 2 3 5 8 9  
     ------------------------------------------------------------------------
      4     2       3     2     2 2     2   4 2 2       3   2      2 2 2 2  
     x x x x  + 3x x x x x  - 2x x x x x , x x x  - 4x x x x x  - x x x x  +
      2 4 7 9     1 2 4 8 9     1 2 5 8 9   3 7 8     1 3 7 8 9    2 3 7 9  
     ------------------------------------------------------------------------
           2     2     2 2 2 2     2       3    2 2 4     3 2 2    2 2 3    
     4x x x x x x  + 3x x x x  - 4x x x x x  + x x x , x x x x  + x x x x  -
       1 2 3 7 8 9     1 3 8 9     1 2 3 8 9    1 2 9   1 3 7 8    2 3 7 9  
     ------------------------------------------------------------------------
       2 2   2         2   2 2     2         2     3   2 2     2 2   3  
     4x x x x x  - 4x x x x x  + 4x x x x x x  + 3x x x x  + 3x x x x  -
       1 3 7 8 9     1 2 3 7 9     1 2 3 7 8 9     1 3 8 9     1 2 7 9  
     ------------------------------------------------------------------------
       3     3   4     2       3   2      2 2     2         2     2  
     4x x x x , x x x x  - 4x x x x x  - x x x x x  + 4x x x x x x  +
       1 2 8 9   3 4 7 8     1 3 4 8 9    2 3 4 7 9     1 2 3 4 8 9  
     ------------------------------------------------------------------------
       2 2     2     2       3    2 2   3     3     2    2 2   2    
     3x x x x x  - 4x x x x x  + x x x x , x x x x x  + x x x x x  -
       1 3 5 8 9     1 2 3 5 9    1 2 6 9   1 3 4 7 8    2 3 4 7 9  
     ------------------------------------------------------------------------
       2 2   2         2       2     2         2     3       2     2 2   3  
     4x x x x x  - 4x x x x x x  + 4x x x x x x  + 3x x x x x  + 3x x x x  -
       1 3 4 8 9     1 2 3 4 7 9     1 2 3 4 8 9     1 3 5 8 9     1 2 4 9  
     ------------------------------------------------------------------------
       3     3       2     2    2 2 2 2    3   2           2            
     4x x x x , x x x x x x  - x x x x  + x x x x x  - 4x x x x x x x  +
       1 2 5 9   1 2 3 4 5 8    1 3 5 8    2 3 4 8 9     1 2 3 4 5 8 9  
     ------------------------------------------------------------------------
       2     2        4 2 2       3     2     2 2 2 2   4 2 2       3        
     3x x x x x x  - x x x  + 3x x x x x  - 2x x x x , x x x  - 4x x x x x x 
       1 2 3 5 8 9    2 4 9     1 2 4 5 9     1 2 5 9   3 4 8     1 3 4 5 8 9
     ------------------------------------------------------------------------
        2 2 2 2         2     2     2 2 2 2     2         2    2 2 2 2 
     - x x x x  + 4x x x x x x  + 3x x x x  - 4x x x x x x  + x x x x ,
        2 3 4 9     1 2 3 4 5 9     1 3 5 9     1 2 3 5 6 9    1 2 6 9 
     ------------------------------------------------------------------------
        3 2 2    2 2 2         2 2               2   2 2     2         2  
     x x x x  + x x x x x  - 4x x x x x x  - 4x x x x x  + 4x x x x x x  +
      1 3 4 8    2 3 4 7 9     1 3 4 5 8 9     1 2 3 4 9     1 2 3 4 5 9  
     ------------------------------------------------------------------------
       3   2 2     2 2     2     3       2       2   2      2 2 3    
     3x x x x  + 3x x x x x  - 4x x x x x , x x x x x x  - x x x x  +
       1 3 5 9     1 2 4 6 9     1 2 5 6 9   1 2 3 4 5 8    1 3 5 8  
     ------------------------------------------------------------------------
      3   2           2     2       2     3      4 2           3          
     x x x x x  - 4x x x x x x  + 3x x x x x  - x x x x  + 3x x x x x x  -
      2 3 4 5 9     1 2 3 4 5 9     1 2 3 5 9    2 4 6 9     1 2 4 5 6 9  
     ------------------------------------------------------------------------
       2 2 2       4 2           3   2      2 2 2             2          
     2x x x x x , x x x x  - 4x x x x x  - x x x x x  + 4x x x x x x x  +
       1 2 5 6 9   3 4 5 8     1 3 4 5 9    2 3 4 6 9     1 2 3 4 5 6 9  
     ------------------------------------------------------------------------
       2 2 2         2       2      2 2 3       3 2        2 2 3    
     3x x x x x  - 4x x x x x x  + x x x x , x x x x x  + x x x x  -
       1 3 5 6 9     1 2 3 5 6 9    1 2 6 9   1 3 4 5 8    2 3 4 9  
     ------------------------------------------------------------------------
       2 2   2         2   2         2                 3   2      
     4x x x x x  - 4x x x x x x  + 4x x x x x x x  + 3x x x x x  +
       1 3 4 5 9     1 2 3 4 6 9     1 2 3 4 5 6 9     1 3 5 6 9  
     ------------------------------------------------------------------------
       2 2   2       3     2     2 2 4    2 2 2 2       2   3    
     3x x x x x  - 4x x x x x , x x x  - x x x x  - 4x x x x x  +
       1 2 4 6 9     1 2 5 6 9   2 3 7    1 3 7 8     1 2 3 7 9  
     ------------------------------------------------------------------------
       2     2         2 2 2 2     3       2    4 2 2   2 2   3    2 2     2
     4x x x x x x  + 3x x x x  - 4x x x x x  + x x x , x x x x  - x x x x x 
       1 2 3 7 8 9     1 2 7 9     1 2 7 8 9    1 8 9   2 3 4 7    1 3 4 7 8
     ------------------------------------------------------------------------
           2     2       2                 2 2     2     3       2  
     - 4x x x x x x  + 4x x x x x x x  + 3x x x x x  - 4x x x x x  +
         1 2 3 4 7 9     1 2 3 4 7 8 9     1 2 4 7 9     1 2 4 8 9  
     ------------------------------------------------------------------------
      4     2   2 2 2 2    2 2 2 2       2   2         2     2      
     x x x x , x x x x  - x x x x  - 4x x x x x x  + 4x x x x x x  +
      1 5 8 9   2 3 4 7    1 3 4 8     1 2 3 4 7 9     1 2 3 4 8 9  
     ------------------------------------------------------------------------
       2 2 2 2     3       2    4 2 2   2 2 3      2 2 2           2   3    
     3x x x x  - 4x x x x x  + x x x , x x x x  - x x x x x  - 4x x x x x  +
       1 2 4 9     1 2 4 5 9    1 5 9   2 3 4 7    1 3 4 5 8     1 2 3 4 9  
     ------------------------------------------------------------------------
       2     2         2 2 2         3              4 2           2   3  
     4x x x x x x  + 3x x x x x  - 4x x x x x x  + x x x x , x x x x x  -
       1 2 3 4 5 9     1 2 4 6 9     1 2 4 5 6 9    1 5 6 9   1 2 3 4 5  
     ------------------------------------------------------------------------
      2 2 4    3   2           2     2       2     3      4 2 2       3     2
     x x x  + x x x x x  - 4x x x x x x  + 3x x x x x  - x x x  + 3x x x x x 
      1 3 5    2 3 4 5 6     1 2 3 4 5 6     1 2 3 5 6    2 4 6     1 2 4 5 6
     ------------------------------------------------------------------------
         2 2 2 2   4 2 2       3   2      2 2 2 2         2     2     2 2 2 2
     - 2x x x x , x x x  - 4x x x x x  - x x x x  + 4x x x x x x  + 3x x x x 
         1 2 5 6   3 4 5     1 3 4 5 6    2 3 4 6     1 2 3 4 5 6     1 3 5 6
     ------------------------------------------------------------------------
         2       3    2 2 4     3 2 2    2 2 3       2 2   2         2   2 2
     - 4x x x x x  + x x x , x x x x  + x x x x  - 4x x x x x  - 4x x x x x 
         1 2 3 5 6    1 2 6   1 3 4 5    2 3 4 6     1 3 4 5 6     1 2 3 4 6
     ------------------------------------------------------------------------
         2         2     3   2 2     2 2   3     3     3   2 2 4    2 2 2 2  
     + 4x x x x x x  + 3x x x x  + 3x x x x  - 4x x x x , x x x  - x x x x  -
         1 2 3 4 5 6     1 3 5 6     1 2 4 6     1 2 5 6   2 3 4    1 3 4 5  
     ------------------------------------------------------------------------
         2   3       2     2         2 2 2 2     3       2    4 2 2
     4x x x x x  + 4x x x x x x  + 3x x x x  - 4x x x x x  + x x x )
       1 2 3 4 6     1 2 3 4 5 6     1 2 4 6     1 2 4 5 6    1 5 6

o5 : Ideal of R
i6 : isPrimary Q

o6 = true

Ways to use joinIdeals :

For the programmer

The object joinIdeals is a method function.