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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                         3                      5 2          
o3 = (map(R,R,{-x  + 2x  + x , x , 3x  + -x  + x , x }), ideal (-x  + 2x x  +
               3 1     2    4   1    1   7 2    3   2           3 1     1 2  
     ------------------------------------------------------------------------
                 3     44 2 2   6   3   2 2           2       2       3   2
     x x  + 1, 2x x  + --x x  + -x x  + -x x x  + 2x x x  + 3x x x  + -x x x 
      1 4        1 2    7 1 2   7 1 2   3 1 2 3     1 2 3     1 2 4   7 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

               10                   1               3     3              
o6 = (map(R,R,{--x  + 3x  + x , x , -x  + 9x  + x , -x  + -x  + x , x }),
                7 1     2    5   1  2 1     2    4  2 1   4 2    3   2   
     ------------------------------------------------------------------------
            10 2                   3  1000 3     900 2 2   300 2      
     ideal (--x  + 3x x  + x x  - x , ----x x  + ---x x  + ---x x x  +
             7 1     1 2    1 5    2   343 1 2    49 1 2    49 1 2 5  
     ------------------------------------------------------------------------
     270   3   180   2     30     2      4      3       2 2      3
     ---x x  + ---x x x  + --x x x  + 27x  + 27x x  + 9x x  + x x ), {x , x ,
      7  1 2    7  1 2 5    7 1 2 5      2      2 5     2 5    2 5     5   4 
     ------------------------------------------------------------------------
     x })
      3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                      
     {-10} | 70x_1x_2x_5^6-5400x_2^9x_5-17010x_2^9+900x_2^8x_5^2+5670x_2^
     {-9}  | 13230x_1x_2^2x_5^3-700x_1x_2x_5^5+4410x_1x_2x_5^4+54000x_2^9
     {-9}  | 472588830x_1x_2^3+25004700x_1x_2^2x_5^2+315059220x_1x_2^2x_5
     {-3}  | 10x_1^2+21x_1x_2+7x_1x_5-7x_2^3                             
     ------------------------------------------------------------------------
                                                                         
     8x_5-100x_2^7x_5^3-1890x_2^7x_5^2+630x_2^6x_5^3-210x_2^5x_5^4+70x_2^
     -9000x_2^8x_5-18900x_2^8+1000x_2^7x_5^2+12600x_2^7x_5-6300x_2^6x_5^2
     +140000x_1x_2x_5^5-441000x_1x_2x_5^4+5556600x_1x_2x_5^3+52509870x_1x
                                                                         
     ------------------------------------------------------------------------
                                                                         
     4x_5^5+147x_2^2x_5^6+49x_2x_5^7                                     
     +2100x_2^5x_5^3-700x_2^4x_5^4+4410x_2^4x_5^3+27783x_2^3x_5^3-1470x_2
     _2x_5^2-10800000x_2^9+1800000x_2^8x_5+5670000x_2^8-200000x_2^7x_5^2-
                                                                         
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ^2x_5^5+18522x_2^2x_5^4-490x_2x_5^6+3087x_2x_5^5                        
     3150000x_2^7x_5+3969000x_2^7+1260000x_2^6x_5^2-3969000x_2^6x_5-25004700x
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     _2^6-420000x_2^5x_5^3+1323000x_2^5x_5^2+8334900x_2^5x_5+157529610x_2^5+
                                                                            
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     140000x_2^4x_5^4-441000x_2^4x_5^3+5556600x_2^4x_5^2+52509870x_2^4x_5+
                                                                          
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     992436543x_2^4+52509870x_2^3x_5^2+992436543x_2^3x_5+294000x_2^2x_5^5-
                                                                          
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     926100x_2^2x_5^4+29172150x_2^2x_5^3+330812181x_2^2x_5^2+98000x_2x_5^6-
                                                                           
     ------------------------------------------------------------------------
                                                     |
                                                     |
                                                     |
     308700x_2x_5^5+3889620x_2x_5^4+36756909x_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,{a + 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

                1                  1     4                      7 2         
o13 = (map(R,R,{-x  + x  + x , x , -x  + -x  + x , x }), ideal (-x  + x x  +
                6 1    2    4   1  2 1   7 2    3   2           6 1    1 2  
      -----------------------------------------------------------------------
                 1 3     25 2 2   4   3   1 2          2     1 2      
      x x  + 1, --x x  + --x x  + -x x  + -x x x  + x x x  + -x x x  +
       1 4      12 1 2   42 1 2   7 1 2   6 1 2 3    1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      4   2
      -x x x  + x x x x  + 1), {x , x })
      7 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

                9     1             9      7                      16 2  
o16 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (--x  +
                7 1   3 2    4   1  5 1   10 2    3   2            7 1  
      -----------------------------------------------------------------------
      1                 81 3     3 2 2    7   3   9 2       1   2     9 2    
      -x x  + x x  + 1, --x x  + -x x  + --x x  + -x x x  + -x x x  + -x x x 
      3 1 2    1 4      35 1 2   2 1 2   30 1 2   7 1 2 3   3 1 2 3   5 1 2 4
      -----------------------------------------------------------------------
         7   2
      + --x x x  + x x x x  + 1), {x , x })
        10 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
--trying with basis element limit: 40
--trying with basis element limit: 60
--trying with basis element limit: 80
--trying with basis element limit: infinity
--trying random transformation: 2
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                   2        
o19 = (map(R,R,{2x  - 2x  + x , x , 2x  - 5x  + x , x }), ideal (3x  - 2x x 
                  1     2    4   1    1     2    3   2             1     1 2
      -----------------------------------------------------------------------
                    3        2 2        3     2           2       2      
      + x x  + 1, 4x x  - 14x x  + 10x x  + 2x x x  - 2x x x  + 2x x x  -
         1 4        1 2      1 2      1 2     1 2 3     1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      5x x x  + x x x x  + 1), {x , x })
        1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :