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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

               10     1             9                            17 2   1    
o3 = (map(R,R,{--x  + -x  + x , x , -x  + 6x  + x , x }), ideal (--x  + -x x 
                7 1   2 2    4   1  7 1     2    3   2            7 1   2 1 2
     ------------------------------------------------------------------------
                 90 3     129 2 2       3   10 2       1   2     9 2      
     + x x  + 1, --x x  + ---x x  + 3x x  + --x x x  + -x x x  + -x x x  +
        1 4      49 1 2    14 1 2     1 2    7 1 2 3   2 1 2 3   7 1 2 4  
     ------------------------------------------------------------------------
         2
     6x 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

               5     3             3                1     10              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + 2x  + x , --x  + --x  + x , x }),
               3 1   8 2    5   1  5 1     2    4  10 1    7 2    3   2   
     ------------------------------------------------------------------------
            5 2   3               3  125 3     25 2 2   25 2       45   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
            3 1   8 1 2    1 5    2   27 1 2    8 1 2    3 1 2 5   64 1 2  
     ------------------------------------------------------------------------
     15   2           2    27 4   27 3     9 2 2      3
     --x x x  + 5x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
      4 1 2 5     1 2 5   512 2   64 2 5   8 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                               
     {-10} | 491520x_1x_2x_5^6-691200x_2^9x_5-3645x_2^9+921600x_2^
     {-9}  | 9720x_1x_2^2x_5^3-2457600x_1x_2x_5^5+25920x_1x_2x_5^4
     {-9}  | 63772920x_1x_2^3+16124313600x_1x_2^2x_5^2+340122240x_
     {-3}  | 40x_1^2+9x_1x_2+24x_1x_5-24x_2^3                     
     ------------------------------------------------------------------------
                                                                
     8x_5^2+9720x_2^8x_5-819200x_2^7x_5^3-25920x_2^7x_5^2+69120x
     +3456000x_2^9-4608000x_2^8x_5-16200x_2^8+4096000x_2^7x_5^2+
     1x_2^2x_5+257698037760000x_1x_2x_5^5-1358954496000x_1x_2x_5
                                                                
     ------------------------------------------------------------------------
                                                 
     _2^6x_5^3-184320x_2^5x_5^4+491520x_2^4x_5^5+
     86400x_2^7x_5-345600x_2^6x_5^2+921600x_2^5x_
     ^4+28665446400x_1x_2x_5^3+453496320x_1x_2x_5
                                                 
     ------------------------------------------------------------------------
                                                                        
     110592x_2^2x_5^6+294912x_2x_5^7                                    
     5^3-2457600x_2^4x_5^4+25920x_2^4x_5^3+2187x_2^3x_5^3-552960x_2^2x_5
     ^2-362387865600000x_2^9+483183820800000x_2^8x_5+2548039680000x_2^8-
                                                                        
     ------------------------------------------------------------------------
                                                                       
                                                                       
     ^5+11664x_2^2x_5^4-1474560x_2x_5^6+15552x_2x_5^5                  
     429496729600000x_2^7x_5^2-11324620800000x_2^7x_5+23887872000x_2^7+
                                                                       
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     36238786560000x_2^6x_5^2-191102976000x_2^6x_5-2015539200x_2^6-
                                                                   
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     96636764160000x_2^5x_5^3+509607936000x_2^5x_5^2+5374771200x_2^5x_5+
                                                                        
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     170061120x_2^5+257698037760000x_2^4x_5^4-1358954496000x_2^4x_5^3+
                                                                      
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     28665446400x_2^4x_5^2+453496320x_2^4x_5+14348907x_2^4+3627970560x_2^3x_5
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     ^2+114791256x_2^3x_5+57982058496000x_2^2x_5^5-305764761600x_2^2x_5^4+
                                                                          
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     16124313600x_2^2x_5^3+306110016x_2^2x_5^2+154618822656000x_2x_5^6-
                                                                       
     ------------------------------------------------------------------------
                                                                |
                                                                |
                                                                |
     815372697600x_2x_5^5+17199267840x_2x_5^4+272097792x_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

                3     1             1     7                      10 2   1    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                7 1   2 2    4   1  2 1   4 2    3   2            7 1   2 1 2
      -----------------------------------------------------------------------
                   3 3      2 2   7   3   3 2       1   2     1 2      
      + x x  + 1, --x x  + x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      14 1 2    1 2   8 1 2   7 1 2 3   2 1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      7   2
      -x x x  + x x x x  + 1), {x , x })
      4 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

                4                  1     4                      7 2         
o16 = (map(R,R,{-x  + x  + x , x , -x  + -x  + x , x }), ideal (-x  + x x  +
                3 1    2    4   1  4 1   5 2    3   2           3 1    1 2  
      -----------------------------------------------------------------------
                1 3     79 2 2   4   3   4 2          2     1 2       4   2
      x x  + 1, -x x  + --x x  + -x x  + -x x x  + x x x  + -x x x  + -x x x 
       1 4      3 1 2   60 1 2   5 1 2   3 1 2 3    1 2 3   4 1 2 4   5 1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         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,{x  - 2x  + x , x , x  - 2x  + x , x }), ideal (2x  - 2x x  +
                 1     2    4   1   1     2    3   2             1     1 2  
      -----------------------------------------------------------------------
                 3       2 2       3    2           2      2           2
      x x  + 1, x x  - 4x x  + 4x x  + x x x  - 2x x x  + x x x  - 2x x x  +
       1 4       1 2     1 2     1 2    1 2 3     1 2 3    1 2 4     1 2 4  
      -----------------------------------------------------------------------
      x x x x  + 1), {x , x })
       1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :