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

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     1                        2   1      
o3 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , x }), ideal (2x  + -x x  +
                1   6 2    4   1  2 1   2 2    3   2             1   6 1 2  
     ------------------------------------------------------------------------
               3 3     3 2 2    1   3    2       1   2     3 2       1   2
     x x  + 1, -x x  + -x x  + --x x  + x x x  + -x x x  + -x x x  + -x x x 
      1 4      2 1 2   4 1 2   12 1 2    1 2 3   6 1 2 3   2 1 2 4   2 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

               9     1                  3               1                    
o6 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , 9x  + -x  + x , x }), ideal
               4 1   4 2    5   1   1   5 2    4    1   2 2    3   2         
     ------------------------------------------------------------------------
      9 2   1               3  729 3     243 2 2   243 2       27   3  
     (-x  + -x x  + x x  - x , ---x x  + ---x x  + ---x x x  + --x x  +
      4 1   4 1 2    1 5    2   64 1 2    64 1 2    16 1 2 5   64 1 2  
     ------------------------------------------------------------------------
     27   2     27     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 })
      8 1 2 5    4 1 2 5   64 2   16 2 5   4 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                    
     {-10} | 9216x_1x_2x_5^6-7776x_2^9x_5-9x_2^9+15552x_2^8x_5^2+36x_2^
     {-9}  | 36x_1x_2^2x_5^3-62208x_1x_2x_5^5+144x_1x_2x_5^4+52488x_2^9
     {-9}  | 9x_1x_2^3+15552x_1x_2^2x_5^2+72x_1x_2^2x_5+7739670528x_1x_
     {-3}  | 9x_1^2+x_1x_2+4x_1x_5-4x_2^3                              
     ------------------------------------------------------------------------
                                                     
     8x_5-20736x_2^7x_5^3-144x_2^7x_5^2+576x_2^6x_5^3
     -104976x_2^8x_5-81x_2^8+139968x_2^7x_5^2+648x_2^
     2x_5^5-8957952x_1x_2x_5^4+41472x_1x_2x_5^3+144x_
                                                     
     ------------------------------------------------------------------------
                                                                
     -2304x_2^5x_5^4+9216x_2^4x_5^5+1024x_2^2x_5^6+4096x_2x_5^7 
     7x_5-3888x_2^6x_5^2+15552x_2^5x_5^3-62208x_2^4x_5^4+144x_2^
     1x_2x_5^2-6530347008x_2^9+13060694016x_2^8x_5+15116544x_2^8
                                                                
     ------------------------------------------------------------------------
                                                                             
                                                                             
     4x_5^3+4x_2^3x_5^3-6912x_2^2x_5^5+32x_2^2x_5^4-27648x_2x_5^6+64x_2x_5^5 
     -17414258688x_2^7x_5^2-100776960x_2^7x_5+46656x_2^7+483729408x_2^6x_5^2-
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     559872x_2^6x_5-1296x_2^6-1934917632x_2^5x_5^3+2239488x_2^5x_5^2+5184x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5x_5+36x_2^5+7739670528x_2^4x_5^4-8957952x_2^4x_5^3+41472x_2^4x_5^2+144x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _2^4x_5+x_2^4+1728x_2^3x_5^2+12x_2^3x_5+859963392x_2^2x_5^5-995328x_2^2x
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     _5^4+11520x_2^2x_5^3+48x_2^2x_5^2+3439853568x_2x_5^6-3981312x_2x_5^5+
                                                                          
     ------------------------------------------------------------------------
                              |
                              |
                              |
     18432x_2x_5^4+64x_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     1                        2          
o13 = (map(R,R,{x  + 8x  + x , x , -x  + -x  + x , x }), ideal (2x  + 8x x  +
                 1     2    4   1  3 1   3 2    3   2             1     1 2  
      -----------------------------------------------------------------------
                5 3     41 2 2   8   3    2           2     5 2       1   2
      x x  + 1, -x x  + --x x  + -x x  + x x x  + 8x x x  + -x x x  + -x x x 
       1 4      3 1 2    3 1 2   3 1 2    1 2 3     1 2 3   3 1 2 4   3 1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         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     8             4                            4 2   8    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + 3x  + x , x }), ideal (-x  + -x x 
                3 1   5 2    4   1  3 1     2    3   2           3 1   5 1 2
      -----------------------------------------------------------------------
                  4 3     47 2 2   24   3   1 2       8   2     4 2      
      + x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      9 1 2   15 1 2    5 1 2   3 1 2 3   5 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
          2
      3x x x  + x x x x  + 1), {x , x })
        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  + 2x  + x , x , - 2x  + x , x }), ideal (- 2x  + 2x x 
                    1     2    4   1      2    3   2               1     1 2
      -----------------------------------------------------------------------
                    2 2       3     2           2         2
      + x x  + 1, 6x x  - 4x x  - 3x x x  + 2x x x  - 2x x x  + x x x x  +
         1 4        1 2     1 2     1 2 3     1 2 3     1 2 4    1 2 3 4  
      -----------------------------------------------------------------------
      1), {x , x })
            4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :