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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

                    2             7     1                        2   2      
o3 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , x }), ideal (2x  + -x x  +
                1   5 2    4   1  8 1   3 2    3   2             1   5 1 2  
     ------------------------------------------------------------------------
               7 3     41 2 2    2   3    2       2   2     7 2       1   2
     x x  + 1, -x x  + --x x  + --x x  + x x x  + -x x x  + -x x x  + -x x x 
      1 4      8 1 2   60 1 2   15 1 2    1 2 3   5 1 2 3   8 1 2 4   3 1 2 4
     ------------------------------------------------------------------------
     + x x x x  + 1), {x , x })
        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     7             8     8               1              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , 9x  + -x  + x , x }),
               4 1   2 2    5   1  5 1   3 2    4    1   2 2    3   2   
     ------------------------------------------------------------------------
            5 2   7               3  125 3     525 2 2   75 2       735   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x  +
            4 1   2 1 2    1 5    2   64 1 2    32 1 2   16 1 2 5    16 1 2  
     ------------------------------------------------------------------------
     105   2     15     2   343 4   147 3     21 2 2      3
     ---x x x  + --x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
      4  1 2 5    4 1 2 5    8  2    4  2 5    2 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 160x_1x_2x_5^6-14700x_2^9x_5-84035x_2^9+2100x_2^8x_5^2+24010x_2^
     {-9}  | 48020x_1x_2^2x_5^3-1200x_1x_2x_5^5+13720x_1x_2x_5^4+110250x_2^9-
     {-9}  | 9886633715x_1x_2^3+247062900x_1x_2^2x_5^2+5649504980x_1x_2^2x_5+
     {-3}  | 5x_1^2+14x_1x_2+4x_1x_5-4x_2^3                                  
     ------------------------------------------------------------------------
                                                                           
     8x_5-200x_2^7x_5^3-6860x_2^7x_5^2+1960x_2^6x_5^3-560x_2^5x_5^4+160x_2^
     15750x_2^8x_5-60025x_2^8+1500x_2^7x_5^2+34300x_2^7x_5-14700x_2^6x_5^2+
     360000x_1x_2x_5^5-2058000x_1x_2x_5^4+47059600x_1x_2x_5^3+807072140x_1x
                                                                           
     ------------------------------------------------------------------------
                                                                          
     4x_5^5+448x_2^2x_5^6+128x_2x_5^7                                     
     4200x_2^5x_5^3-1200x_2^4x_5^4+13720x_2^4x_5^3+134456x_2^3x_5^3-3360x_
     _2x_5^2-33075000x_2^9+4725000x_2^8x_5+27011250x_2^8-450000x_2^7x_5^2-
                                                                          
     ------------------------------------------------------------------------
                                                                       
                                                                       
     2^2x_5^5+76832x_2^2x_5^4-960x_2x_5^6+10976x_2x_5^5                
     12862500x_2^7x_5+29412250x_2^7+4410000x_2^6x_5^2-25210500x_2^6x_5-
                                                                       
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     288240050x_2^6-1260000x_2^5x_5^3+7203000x_2^5x_5^2+82354300x_2^5x_5+
                                                                         
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     2824752490x_2^5+360000x_2^4x_5^4-2058000x_2^4x_5^3+47059600x_2^4x_5^2+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     807072140x_2^4x_5+27682574402x_2^4+691776120x_2^3x_5^2+23727920916x_2^3x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5+1008000x_2^2x_5^5-5762400x_2^2x_5^4+329417200x_2^2x_5^3+6779405976x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2x_5^2+288000x_2x_5^6-1646400x_2x_5^5+37647680x_2x_5^4+645657712x_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

                2     1                                          5 2   1    
o13 = (map(R,R,{-x  + -x  + x , x , 9x  + 2x  + x , x }), ideal (-x  + -x x 
                3 1   7 2    4   1    1     2    3   2           3 1   7 1 2
      -----------------------------------------------------------------------
                    3     55 2 2   2   3   2 2       1   2       2      
      + x x  + 1, 6x x  + --x x  + -x x  + -x x x  + -x x x  + 9x x x  +
         1 4        1 2   21 1 2   7 1 2   3 1 2 3   7 1 2 3     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                     7                            14 2  
o16 = (map(R,R,{-x  + 10x  + x , x , --x  + 2x  + x , x }), ideal (--x  +
                9 1      2    4   1  10 1     2    3   2            9 1  
      -----------------------------------------------------------------------
                          7 3     73 2 2        3   5 2            2    
      10x x  + x x  + 1, --x x  + --x x  + 20x x  + -x x x  + 10x x x  +
         1 2    1 4      18 1 2    9 1 2      1 2   9 1 2 3      1 2 3  
      -----------------------------------------------------------------------
       7 2           2
      --x x x  + 2x x x  + x x x x  + 1), {x , x })
      10 1 2 4     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

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

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :