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NoetherNormalization :: noetherNormalization

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

                     5             7     1                        2   5      
o3 = (map(R,R,{2x  + -x  + x , x , -x  + -x  + x , x }), ideal (3x  + -x x  +
                 1   3 2    4   1  4 1   4 2    3   2             1   3 1 2  
     ------------------------------------------------------------------------
               7 3     41 2 2    5   3     2       5   2     7 2      
     x x  + 1, -x x  + --x x  + --x x  + 2x x x  + -x x x  + -x x x  +
      1 4      2 1 2   12 1 2   12 1 2     1 2 3   3 1 2 3   4 1 2 4  
     ------------------------------------------------------------------------
     1   2
     -x x x  + x x x x  + 1), {x , x })
     4 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     1             1     1         8                         
o6 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , -x  + x  + x , x }), ideal
               8 1   8 2    5   1  2 1   2 2    4  9 1    2    3   2         
     ------------------------------------------------------------------------
      5 2   1               3  125 3      75 2 2   75 2        15   3  
     (-x  + -x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x  +
      8 1   8 1 2    1 5    2  512 1 2   512 1 2   64 1 2 5   512 1 2  
     ------------------------------------------------------------------------
     15   2     15     2    1  4    3 3     3 2 2      3
     --x x x  + --x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
     32 1 2 5    8 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} | 163840x_1x_2x_5^6-9600x_2^9x_5-5x_2^9+38400x_2^8x_5^2+40x_2^
     {-9}  | 40x_1x_2^2x_5^3-307200x_1x_2x_5^5+320x_1x_2x_5^4+18000x_2^9-
     {-9}  | 5x_1x_2^3+38400x_1x_2^2x_5^2+80x_1x_2^2x_5+188743680000x_1x_
     {-3}  | 5x_1^2+x_1x_2+8x_1x_5-8x_2^3                                
     ------------------------------------------------------------------------
                                                                
     8x_5-102400x_2^7x_5^3-320x_2^7x_5^2+2560x_2^6x_5^3-20480x_2
     72000x_2^8x_5-25x_2^8+192000x_2^7x_5^2+400x_2^7x_5-4800x_2^
     2x_5^5-98304000x_1x_2x_5^4+204800x_1x_2x_5^3+320x_1x_2x_5^2
                                                                
     ------------------------------------------------------------------------
                                                                             
     ^5x_5^4+163840x_2^4x_5^5+32768x_2^2x_5^6+262144x_2x_5^7                 
     6x_5^2+38400x_2^5x_5^3-307200x_2^4x_5^4+320x_2^4x_5^3+8x_2^3x_5^3-61440x
     -11059200000x_2^9+44236800000x_2^8x_5+23040000x_2^8-117964800000x_2^7x_5
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
     _2^2x_5^5+128x_2^2x_5^4-491520x_2x_5^6+512x_2x_5^5                   
     ^2-307200000x_2^7x_5+64000x_2^7+2949120000x_2^6x_5^2-1536000x_2^6x_5-
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1600x_2^6-23592960000x_2^5x_5^3+12288000x_2^5x_5^2+12800x_2^5x_5+40x_2^5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     +188743680000x_2^4x_5^4-98304000x_2^4x_5^3+204800x_2^4x_5^2+320x_2^4x_5+
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     x_2^4+7680x_2^3x_5^2+24x_2^3x_5+37748736000x_2^2x_5^5-19660800x_2^2x_5^4
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     +102400x_2^2x_5^3+192x_2^2x_5^2+301989888000x_2x_5^6-157286400x_2x_5^5+
                                                                            
     ------------------------------------------------------------------------
                                |
                                |
                                |
     327680x_2x_5^4+512x_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

                2     5                                          9 2   5    
o13 = (map(R,R,{-x  + -x  + x , x , 2x  + 8x  + x , x }), ideal (-x  + -x x 
                7 1   9 2    4   1    1     2    3   2           7 1   9 1 2
      -----------------------------------------------------------------------
                  4 3     214 2 2   40   3   2 2       5   2       2      
      + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + 2x x x  +
         1 4      7 1 2    63 1 2    9 1 2   7 1 2 3   9 1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      8x 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

                1     5              1     1                      3 2   5    
o16 = (map(R,R,{-x  + -x  + x , x , --x  + -x  + x , x }), ideal (-x  + -x x 
                2 1   2 2    4   1  10 1   2 2    3   2           2 1   2 1 2
      -----------------------------------------------------------------------
                   1 3     1 2 2   5   3   1 2       5   2      1 2      
      + x x  + 1, --x x  + -x x  + -x x  + -x x x  + -x x x  + --x x x  +
         1 4      20 1 2   2 1 2   4 1 2   2 1 2 3   2 1 2 3   10 1 2 4  
      -----------------------------------------------------------------------
      1   2
      -x x x  + x x x x  + 1), {x , x })
      2 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,{3x  + x , x , - 4x  + x  + x , x }), ideal (x  + 3x x  + x x 
                  2    4   1      1    2    3   2            1     1 2    1 4
      -----------------------------------------------------------------------
                2 2       3       2       2          2
      + 1, - 12x x  + 3x x  + 3x x x  - 4x x x  + x x x  + x x x x  + 1),
                1 2     1 2     1 2 3     1 2 4    1 2 4    1 2 3 4      
      -----------------------------------------------------------------------
      {x , x })
        4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :