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noetherNormalization -- data for Noether normalization

Synopsis

Description

The computations performed in the routine noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
i1 : R = QQ[x_1..x_4];
i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o2 : Ideal of R
i3 : (f,J,X) = noetherNormalization I

               1     9             4                            6 2   9      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + 4x  + x , x }), ideal (-x  + -x x  +
               5 1   8 2    4   1  5 1     2    3   2           5 1   8 1 2  
     ------------------------------------------------------------------------
                4 3     17 2 2   9   3   1 2       9   2     4 2      
     x x  + 1, --x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  +
      1 4      25 1 2   10 1 2   2 1 2   5 1 2 3   8 1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
         2
     4x x x  + x x x x  + 1), {x , x })
       1 2 4    1 2 3 4         4   3

o3 : Sequence
The next example shows how when we use the lexicographical ordering, we can see the integrality of R/ f I over the polynomial ring in dim(R/I) variables:
i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];
i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);

o5 : Ideal of R
i6 : (f,J,X) = noetherNormalization I

               9     4                   7         7     4              
o6 = (map(R,R,{-x  + -x  + x , x , 4x  + -x  + x , -x  + -x  + x , x }),
               7 1   5 2    5   1    1   5 2    4  8 1   3 2    3   2   
     ------------------------------------------------------------------------
            9 2   4               3  729 3     972 2 2   243 2       432   3
     ideal (-x  + -x x  + x x  - x , ---x x  + ---x x  + ---x x x  + ---x x 
            7 1   5 1 2    1 5    2  343 1 2   245 1 2    49 1 2 5   175 1 2
     ------------------------------------------------------------------------
       216   2     27     2    64 4   48 3     12 2 2      3
     + ---x x x  + --x x x  + ---x  + --x x  + --x x  + x x ), {x , x , x })
        35 1 2 5    7 1 2 5   125 2   25 2 5    5 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                 
     {-10} | 196875x_1x_2x_5^6-972000x_2^9x_5-64512x
     {-9}  | 564480x_1x_2^2x_5^3-5315625x_1x_2x_5^5+
     {-9}  | 226586787840x_1x_2^3+2133734400000x_1x_
     {-3}  | 45x_1^2+28x_1x_2+35x_1x_5-35x_2^3      
     ------------------------------------------------------------------------
                                                                 
     _2^9+607500x_2^8x_5^2+80640x_2^8x_5-253125x_2^7x_5^3-100800x
     705600x_1x_2x_5^4+26244000x_2^9-16402500x_2^8x_5-725760x_2^8
     2^2x_5^2+566466969600x_1x_2^2x_5+100913818359375x_1x_2x_5^5-
                                                                 
     ------------------------------------------------------------------------
                                                                            
     _2^7x_5^2+126000x_2^6x_5^3-157500x_2^5x_5^4+196875x_2^4x_5^5+122500x_2^
     +6834375x_2^7x_5^2+1814400x_2^7x_5-3402000x_2^6x_5^2+4252500x_2^5x_5^3-
     6697687500000x_1x_2x_5^4+1778112000000x_1x_2x_5^3+354041856000x_1x_2x_5
                                                                            
     ------------------------------------------------------------------------
                                                                         
     2x_5^6+153125x_2x_5^7                                               
     5315625x_2^4x_5^4+705600x_2^4x_5^3+351232x_2^3x_5^3-3307500x_2^2x_5^
     ^2-498225937500000x_2^9+311391210937500x_2^8x_5+20667150000000x_2^8-
                                                                         
     ------------------------------------------------------------------------
                                                                         
                                                                         
     5+878080x_2^2x_5^4-4134375x_2x_5^6+548800x_2x_5^5                   
     129746337890625x_2^7x_5^2-43056562500000x_2^7x_5+1143072000000x_2^7+
                                                                         
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     64584843750000x_2^6x_5^2-4286520000000x_2^6x_5-568995840000x_2^6-
                                                                      
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     80731054687500x_2^5x_5^3+5358150000000x_2^5x_5^2+711244800000x_2^5x_5+
                                                                           
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     283233484800x_2^5+100913818359375x_2^4x_5^4-6697687500000x_2^4x_5^3+
                                                                         
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     1778112000000x_2^4x_5^2+354041856000x_2^4x_5+140987334656x_2^4+
                                                                    
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     1327656960000x_2^3x_5^2+528702504960x_2^3x_5+62790820312500x_2^2x_5^5-
                                                                           
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     4167450000000x_2^2x_5^4+2765952000000x_2^2x_5^3+660878131200x_2^2x_5^2+
                                                                            
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     78488525390625x_2x_5^6-5209312500000x_2x_5^5+1382976000000x_2x_5^4+
                                                                        
     ------------------------------------------------------------------------
                          |
                          |
                          |
     275365888000x_2x_5^3 |
                          |

             5       1
o7 : Matrix R  <--- R
If noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
i8 : R = ZZ/2[a,b];
i9 : I = ideal(a^2*b+a*b^2+1);

o9 : Ideal of R
i10 : (f,J,X) = noetherNormalization I
--warning: no good linear transformation found by noetherNormalization

                               2       2
o10 = (map(R,R,{b, a}), ideal(a b + a*b  + 1), {b})

o10 : Sequence
Here is an example with the option Verbose => true:
i11 : R = QQ[x_1..x_4];
i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o12 : Ideal of R
i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20

                      9             1                              2   9    
o13 = (map(R,R,{2x  + -x  + x , x , -x  + 2x  + x , x }), ideal (3x  + -x x 
                  1   5 2    4   1  4 1     2    3   2             1   5 1 2
      -----------------------------------------------------------------------
                  1 3     89 2 2   18   3     2       9   2     1 2      
      + x x  + 1, -x x  + --x x  + --x x  + 2x x x  + -x x x  + -x x x  +
         1 4      2 1 2   20 1 2    5 1 2     1 2 3   5 1 2 3   4 1 2 4  
      -----------------------------------------------------------------------
          2
      2x x x  + x x x x  + 1), {x , x })
        1 2 4    1 2 3 4         4   3

o13 : Sequence
The first number in the output above gives the number of linear transformations performed by the routine while attempting to place I into the desired position. The second number tells which BasisElementLimit was used when computing the (partial) Groebner basis. By default, noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option BasisElementLimit set to predetermined values. The default values come from the following list:{5,20,40,60,80,infinity}. To set the values manually, use the option LimitList:
i14 : R = QQ[x_1..x_4]; 
i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o15 : Ideal of R
i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10})
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 10

                      5             5     9                        2   5    
o16 = (map(R,R,{8x  + -x  + x , x , -x  + -x  + x , x }), ideal (9x  + -x x 
                  1   7 2    4   1  2 1   8 2    3   2             1   7 1 2
      -----------------------------------------------------------------------
                     3     151 2 2   45   3     2       5   2     5 2      
      + x x  + 1, 20x x  + ---x x  + --x x  + 8x x x  + -x x x  + -x x x  +
         1 4         1 2    14 1 2   56 1 2     1 2 3   7 1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      9   2
      -x x x  + x x x x  + 1), {x , x })
      8 1 2 4    1 2 3 4         4   3

o16 : Sequence
To limit the randomness of the coefficients, use the option RandomRange. Here is an example where the coefficients of the linear transformation are random integers from -2 to 2:
i17 : R = QQ[x_1..x_4];
i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o18 : Ideal of R
i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                          2                 
o19 = (map(R,R,{2x  + x , x , x  - x  + x , x }), ideal (x  + 2x x  + x x  +
                  2    4   1   1    2    3   2            1     1 2    1 4  
      -----------------------------------------------------------------------
           2 2       3       2      2          2
      1, 2x x  - 2x x  + 2x x x  + x x x  - x x x  + x x x x  + 1), {x , x })
           1 2     1 2     1 2 3    1 2 4    1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :