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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

               4     1             4     1                      11 2   1    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
               7 1   2 2    4   1  7 1   3 2    3   2            7 1   2 1 2
     ------------------------------------------------------------------------
                 16 3     10 2 2   1   3   4 2       1   2     4 2      
     + x x  + 1, --x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  +
        1 4      49 1 2   21 1 2   6 1 2   7 1 2 3   2 1 2 3   7 1 2 4  
     ------------------------------------------------------------------------
     1   2
     -x x x  + x x x x  + 1), {x , x })
     3 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             7                                         
o6 = (map(R,R,{5x  + -x  + x , x , -x  + 10x  + x , x  + x  + x , x }), ideal
                 1   3 2    5   1  3 1      2    4   1    2    3   2         
     ------------------------------------------------------------------------
        2   5               3      3         2 2      2       125   3  
     (5x  + -x x  + x x  - x , 125x x  + 125x x  + 75x x x  + ---x x  +
        1   3 1 2    1 5    2      1 2       1 2      1 2 5    3  1 2  
     ------------------------------------------------------------------------
          2            2   125 4   25 3       2 2      3
     50x x x  + 15x x x  + ---x  + --x x  + 5x x  + x x ), {x , x , x })
        1 2 5      1 2 5    27 2    3 2 5     2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                   
     {-10} | 1215x_1x_2x_5^6-101250x_2^9x_5-15625x_2^9+30375x_2^8x_5^2
     {-9}  | 1875x_1x_2^2x_5^3-3645x_1x_2x_5^5+1125x_1x_2x_5^4+303750x
     {-9}  | 5859375x_1x_2^3+11390625x_1x_2^2x_5^2+7031250x_1x_2^2x_5+
     {-3}  | 15x_1^2+5x_1x_2+3x_1x_5-3x_2^3                           
     ------------------------------------------------------------------------
                                                                             
     +9375x_2^8x_5-6075x_2^7x_5^3-5625x_2^7x_5^2+3375x_2^6x_5^3-2025x_2^5x_5^
     _2^9-91125x_2^8x_5-9375x_2^8+18225x_2^7x_5^2+11250x_2^7x_5-10125x_2^6x_5
     47829690x_1x_2x_5^5-7381125x_1x_2x_5^4+4556250x_1x_2x_5^3+2109375x_1x_2x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     4+1215x_2^4x_5^5+405x_2^2x_5^6+243x_2x_5^7                              
     ^2+6075x_2^5x_5^3-3645x_2^4x_5^4+1125x_2^4x_5^3+625x_2^3x_5^3-1215x_2^2x
     _5^2-3985807500x_2^9+1195742250x_2^8x_5+184528125x_2^8-239148450x_2^7x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     _5^5+750x_2^2x_5^4-729x_2x_5^6+225x_2x_5^5                              
     ^2-184528125x_2^7x_5+11390625x_2^7+132860250x_2^6x_5^2-20503125x_2^6x_5-
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     6328125x_2^6-79716150x_2^5x_5^3+12301875x_2^5x_5^2+3796875x_2^5x_5+
                                                                        
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     3515625x_2^5+47829690x_2^4x_5^4-7381125x_2^4x_5^3+4556250x_2^4x_5^2+
                                                                         
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2109375x_2^4x_5+1953125x_2^4+3796875x_2^3x_5^2+3515625x_2^3x_5+15943230x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _2^2x_5^5-2460375x_2^2x_5^4+3796875x_2^2x_5^3+2109375x_2^2x_5^2+9565938x
                                                                             
     ------------------------------------------------------------------------
                                                           |
                                                           |
                                                           |
     _2x_5^6-1476225x_2x_5^5+911250x_2x_5^4+421875x_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

                                          4                        2         
o13 = (map(R,R,{2x  + 10x  + x , x , x  + -x  + x , x }), ideal (3x  + 10x x 
                  1      2    4   1   1   9 2    3   2             1      1 2
      -----------------------------------------------------------------------
                    3     98 2 2   40   3     2            2      2      
      + x x  + 1, 2x x  + --x x  + --x x  + 2x x x  + 10x x x  + x x x  +
         1 4        1 2    9 1 2    9 1 2     1 2 3      1 2 3    1 2 4  
      -----------------------------------------------------------------------
      4   2
      -x x x  + x x x x  + 1), {x , x })
      9 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

                     3                                           2   3      
o16 = (map(R,R,{x  + -x  + x , x , 4x  + x  + x , x }), ideal (2x  + -x x  +
                 1   5 2    4   1    1    2    3   2             1   5 1 2  
      -----------------------------------------------------------------------
                  3     17 2 2   3   3    2       3   2       2          2
      x x  + 1, 4x x  + --x x  + -x x  + x x x  + -x x x  + 4x x x  + x x x 
       1 4        1 2    5 1 2   5 1 2    1 2 3   5 1 2 3     1 2 4    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
--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  + 2x  + x , x , 6x  + 2x  + x , x }), ideal (- 2x  +
                    1     2    4   1    1     2    3   2               1  
      -----------------------------------------------------------------------
                             3       2 2       3     2           2    
      2x x  + x x  + 1, - 18x x  + 6x x  + 4x x  - 3x x x  + 2x x x  +
        1 2    1 4           1 2     1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      6x x x  + 2x x x  + x x x x  + 1), {x , x })
        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 :