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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

               10                        5                      19 2        
o3 = (map(R,R,{--x  + 3x  + x , x , x  + -x  + x , x }), ideal (--x  + 3x x 
                9 1     2    4   1   1   6 2    3   2            9 1     1 2
     ------------------------------------------------------------------------
                 10 3     106 2 2   5   3   10 2           2      2      
     + x x  + 1, --x x  + ---x x  + -x x  + --x x x  + 3x x x  + x x x  +
        1 4       9 1 2    27 1 2   2 1 2    9 1 2 3     1 2 3    1 2 4  
     ------------------------------------------------------------------------
     5   2
     -x x x  + x x x x  + 1), {x , x })
     6 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             10     3               5              
o6 = (map(R,R,{-x  + -x  + x , x , --x  + -x  + x , 3x  + -x  + x , x }),
               8 1   3 2    5   1   7 1   4 2    4    1   2 2    3   2   
     ------------------------------------------------------------------------
            5 2   1               3  125 3     25 2 2   75 2        5   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
            8 1   3 1 2    1 5    2  512 1 2   64 1 2   64 1 2 5   24 1 2  
     ------------------------------------------------------------------------
     5   2     15     2    1 4   1 3      2 2      3
     -x x x  + --x x x  + --x  + -x x  + x x  + x x ), {x , x , x })
     4 1 2 5    8 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} | 9720x_1x_2x_5^6-4050x_2^9x_5-40x_2^9+6075x_2^8x_5^2
     {-9}  | 960x_1x_2^2x_5^3-145800x_1x_2x_5^5+2880x_1x_2x_5^4+
     {-9}  | 30720x_1x_2^3+4665600x_1x_2^2x_5^2+184320x_1x_2^2x_
     {-3}  | 15x_1^2+8x_1x_2+24x_1x_5-24x_2^3                   
     ------------------------------------------------------------------------
                                                                             
     +120x_2^8x_5-6075x_2^7x_5^3-360x_2^7x_5^2+1080x_2^6x_5^3-3240x_2^5x_5^4+
     60750x_2^9-91125x_2^8x_5-600x_2^8+91125x_2^7x_5^2+3600x_2^7x_5-16200x_2^
     5+23914845000x_1x_2x_5^5-236196000x_1x_2x_5^4+9331200x_1x_2x_5^3+276480x
                                                                             
     ------------------------------------------------------------------------
                                                                   
     9720x_2^4x_5^5+5184x_2^2x_5^6+15552x_2x_5^7                   
     6x_5^2+48600x_2^5x_5^3-145800x_2^4x_5^4+2880x_2^4x_5^3+512x_2^
     _1x_2x_5^2-9964518750x_2^9+14946778125x_2^8x_5+147622500x_2^8-
                                                                   
     ------------------------------------------------------------------------
                                                                             
                                                                             
     3x_5^3-77760x_2^2x_5^5+3072x_2^2x_5^4-233280x_2x_5^6+4608x_2x_5^5       
     14946778125x_2^7x_5^2-738112500x_2^7x_5+2916000x_2^7+2657205000x_2^6x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     2-26244000x_2^6x_5-518400x_2^6-7971615000x_2^5x_5^3+78732000x_2^5x_5^2+
                                                                            
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     1555200x_2^5x_5+92160x_2^5+23914845000x_2^4x_5^4-236196000x_2^4x_5^3+
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     9331200x_2^4x_5^2+276480x_2^4x_5+16384x_2^4+2488320x_2^3x_5^2+147456x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     3x_5+12754584000x_2^2x_5^5-125971200x_2^2x_5^4+12441600x_2^2x_5^3+
                                                                       
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     442368x_2^2x_5^2+38263752000x_2x_5^6-377913600x_2x_5^5+14929920x_2x_5^4+
                                                                             
     ------------------------------------------------------------------------
                    |
                    |
                    |
     442368x_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             7     1                        2   3    
o13 = (map(R,R,{4x  + -x  + x , x , -x  + -x  + x , x }), ideal (5x  + -x x 
                  1   4 2    4   1  4 1   9 2    3   2             1   4 1 2
      -----------------------------------------------------------------------
                    3     253 2 2    1   3     2       3   2     7 2      
      + x x  + 1, 7x x  + ---x x  + --x x  + 4x x x  + -x x x  + -x x x  +
         1 4        1 2   144 1 2   12 1 2     1 2 3   4 1 2 3   4 1 2 4  
      -----------------------------------------------------------------------
      1   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

                4     7             8     4                      11 2   7    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                7 1   2 2    4   1  5 1   7 2    3   2            7 1   2 1 2
      -----------------------------------------------------------------------
                  32 3     1452 2 2       3   4 2       7   2     8 2      
      + x x  + 1, --x x  + ----x x  + 2x x  + -x x x  + -x x x  + -x x x  +
         1 4      35 1 2    245 1 2     1 2   7 1 2 3   2 1 2 3   5 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

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,{2x  - x  + x , x , - x  + 2x  + x , x }), ideal (3x  - x x  +
                  1    2    4   1     1     2    3   2             1    1 2  
      -----------------------------------------------------------------------
                    3       2 2       3     2          2      2           2
      x x  + 1, - 2x x  + 5x x  - 2x x  + 2x x x  - x 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 :