next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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

               1                   3                            3 2          
o3 = (map(R,R,{-x  + 7x  + x , x , -x  + 2x  + x , x }), ideal (-x  + 7x x  +
               2 1     2    4   1  5 1     2    3   2           2 1     1 2  
     ------------------------------------------------------------------------
                3 3     26 2 2        3   1 2           2     3 2      
     x x  + 1, --x x  + --x x  + 14x x  + -x x x  + 7x x x  + -x x x  +
      1 4      10 1 2    5 1 2      1 2   2 1 2 3     1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
         2
     2x x x  + x x x x  + 1), {x , x })
       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

               3     3             1               10     9              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + 5x  + x , --x  + -x  + x , x }),
               8 1   5 2    5   1  7 1     2    4   3 1   4 2    3   2   
     ------------------------------------------------------------------------
            3 2   3               3   27 3      81 2 2   27 2        81   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x  +
            8 1   5 1 2    1 5    2  512 1 2   320 1 2   64 1 2 5   200 1 2  
     ------------------------------------------------------------------------
     27   2     9     2    27 4   27 3     9 2 2      3
     --x x x  + -x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
     20 1 2 5   8 1 2 5   125 2   25 2 5   5 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                              
     {-10} | 75000x_1x_2x_5^6-60750x_2^9x_5-5832x_2^9+50625x_2^8x
     {-9}  | 8640x_1x_2^2x_5^3-75000x_1x_2x_5^5+14400x_1x_2x_5^4+
     {-9}  | 22394880x_1x_2^3+194400000x_1x_2^2x_5^2+74649600x_1x
     {-3}  | 15x_1^2+24x_1x_2+40x_1x_5-40x_2^3                   
     ------------------------------------------------------------------------
                                                                             
     _5^2+9720x_2^8x_5-28125x_2^7x_5^3-16200x_2^7x_5^2+27000x_2^6x_5^3-45000x
     60750x_2^9-50625x_2^8x_5-3240x_2^8+28125x_2^7x_5^2+10800x_2^7x_5-27000x_
     _2^2x_5+5859375000x_1x_2x_5^5-562500000x_1x_2x_5^4+216000000x_1x_2x_5^3+
                                                                             
     ------------------------------------------------------------------------
                                                                           
     _2^5x_5^4+75000x_2^4x_5^5+120000x_2^2x_5^6+200000x_2x_5^7             
     2^6x_5^2+45000x_2^5x_5^3-75000x_2^4x_5^4+14400x_2^4x_5^3+13824x_2^3x_5
     62208000x_1x_2x_5^2-4746093750x_2^9+3955078125x_2^8x_5+379687500x_2^8-
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ^3-120000x_2^2x_5^5+46080x_2^2x_5^4-200000x_2x_5^6+38400x_2x_5^5        
     2197265625x_2^7x_5^2-1054687500x_2^7x_5+40500000x_2^7+2109375000x_2^6x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2-202500000x_2^6x_5-38880000x_2^6-3515625000x_2^5x_5^3+337500000x_2^5x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5^2+64800000x_2^5x_5+37324800x_2^5+5859375000x_2^4x_5^4-562500000x_2^4x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5^3+216000000x_2^4x_5^2+62208000x_2^4x_5+35831808x_2^4+311040000x_2^3x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2+179159040x_2^3x_5+9375000000x_2^2x_5^5-900000000x_2^2x_5^4+864000000x
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     _2^2x_5^3+298598400x_2^2x_5^2+15625000000x_2x_5^6-1500000000x_2x_5^5+
                                                                          
     ------------------------------------------------------------------------
                                         |
                                         |
                                         |
     576000000x_2x_5^4+165888000x_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

                5                   6     7                      7 2        
o13 = (map(R,R,{-x  + 3x  + x , x , -x  + -x  + x , x }), ideal (-x  + 3x x 
                2 1     2    4   1  5 1   6 2    3   2           2 1     1 2
      -----------------------------------------------------------------------
                    3     391 2 2   7   3   5 2           2     6 2      
      + x x  + 1, 3x x  + ---x x  + -x x  + -x x x  + 3x x x  + -x x x  +
         1 4        1 2    60 1 2   2 1 2   2 1 2 3     1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      7   2
      -x x x  + x x x x  + 1), {x , x })
      6 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

                7     9             10     1                      11 2  
o16 = (map(R,R,{-x  + -x  + x , x , --x  + -x  + x , x }), ideal (--x  +
                4 1   5 2    4   1   3 1   2 2    3   2            4 1  
      -----------------------------------------------------------------------
      9                 35 3     55 2 2    9   3   7 2       9   2    
      -x x  + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  +
      5 1 2    1 4       6 1 2    8 1 2   10 1 2   4 1 2 3   5 1 2 3  
      -----------------------------------------------------------------------
      10 2       1   2
      --x x x  + -x x x  + x x x x  + 1), {x , x })
       3 1 2 4   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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                2          
o19 = (map(R,R,{- 3x  + x , x , - 3x  + 3x  + x , x }), ideal (x  - 3x x  +
                    2    4   1      1     2    3   2            1     1 2  
      -----------------------------------------------------------------------
                  2 2       3       2       2           2
      x x  + 1, 9x x  - 9x x  - 3x x x  - 3x x x  + 3x x x  + x x x x  + 1),
       1 4        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 :