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     8             5                           3 2   8      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , x }), ideal (-x  + -x x  +
               2 1   9 2    4   1  2 1    2    3   2           2 1   9 1 2  
     ------------------------------------------------------------------------
               5 3     49 2 2   8   3   1 2       8   2     5 2          2
     x x  + 1, -x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  + x x x 
      1 4      4 1 2   18 1 2   9 1 2   2 1 2 3   9 1 2 3   2 1 2 4    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

                     7             9     9         1     9              
o6 = (map(R,R,{4x  + -x  + x , x , -x  + -x  + x , -x  + -x  + x , x }),
                 1   8 2    5   1  8 1   4 2    4  9 1   4 2    3   2   
     ------------------------------------------------------------------------
              2   7               3     3        2 2      2       147   3  
     ideal (4x  + -x x  + x x  - x , 64x x  + 42x x  + 48x x x  + ---x x  +
              1   8 1 2    1 5    2     1 2      1 2      1 2 5    16 1 2  
     ------------------------------------------------------------------------
          2            2   343 4   147 3     21 2 2      3
     21x x x  + 12x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
        1 2 5      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} | 32768x_1x_2x_5^6-602112x_2^9x_5-16807x
     {-9}  | 76832x_1x_2^2x_5^3-1572864x_1x_2x_5^5+
     {-9}  | 63274455776x_1x_2^3+1295321137152x_1x_
     {-3}  | 32x_1^2+7x_1x_2+8x_1x_5-8x_2^3        
     ------------------------------------------------------------------------
                                                                 
     _2^9+344064x_2^8x_5^2+19208x_2^8x_5-131072x_2^7x_5^3-21952x_
     87808x_1x_2x_5^4+28901376x_2^9-16515072x_2^8x_5-307328x_2^8+
     2^2x_5^2+144627327488x_1x_2^2x_5+316659348799488x_1x_2x_5^5-
                                                                 
     ------------------------------------------------------------------------
                                                                          
     2^7x_5^2+25088x_2^6x_5^3-28672x_2^5x_5^4+32768x_2^4x_5^5+7168x_2^2x_5
     6291456x_2^7x_5^2+702464x_2^7x_5-1204224x_2^6x_5^2+1376256x_2^5x_5^3-
     8839042695168x_1x_2x_5^4+986911342592x_1x_2x_5^3+82644187136x_1x_2x_5
                                                                          
     ------------------------------------------------------------------------
                                                                        
     ^6+8192x_2x_5^7                                                    
     1572864x_2^4x_5^4+87808x_2^4x_5^3+16807x_2^3x_5^3-344064x_2^2x_5^5+
     ^2-5818615534190592x_2^9+3324923162394624x_2^8x_5+92809948299264x_2
                                                                        
     ------------------------------------------------------------------------
                                                                             
                                                                             
     38416x_2^2x_5^4-393216x_2x_5^6+21952x_2x_5^5                            
     ^8-1266637395197952x_2^7x_5^2-176780853903360x_2^7x_5+1973822685184x_2^7
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     +242442313924608x_2^6x_5^2-6767392063488x_2^6x_5-377801998336x_2^6-
                                                                        
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     277076930199552x_2^5x_5^3+7734162358272x_2^5x_5^2+431773712384x_2^5x_5+
                                                                            
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     72313663744x_2^5+316659348799488x_2^4x_5^4-8839042695168x_2^4x_5^3+
                                                                        
     ------------------------------------------------------------------------
                                                                 
                                                                 
                                                                 
     986911342592x_2^4x_5^2+82644187136x_2^4x_5+13841287201x_2^4+
                                                                 
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     283351498752x_2^3x_5^2+47455841832x_2^3x_5+69269232549888x_2^2x_5^5-
                                                                         
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     1933540589568x_2^2x_5^4+539717140480x_2^2x_5^3+54235247808x_2^2x_5^2+
                                                                          
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     79164837199872x_2x_5^6-2209760673792x_2x_5^5+246727835648x_2x_5^4+
                                                                       
     ------------------------------------------------------------------------
                         |
                         |
                         |
     20661046784x_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

                      3                                          2   3      
o13 = (map(R,R,{2x  + -x  + x , x , x  + x  + x , x }), ideal (3x  + -x x  +
                  1   4 2    4   1   1    2    3   2             1   4 1 2  
      -----------------------------------------------------------------------
                  3     11 2 2   3   3     2       3   2      2          2
      x x  + 1, 2x x  + --x x  + -x x  + 2x x x  + -x x x  + x x x  + x x x 
       1 4        1 2    4 1 2   4 1 2     1 2 3   4 1 2 3    1 2 4    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

                3                                                11 2  
o16 = (map(R,R,{-x  + 10x  + x , x , x  + 5x  + x , x }), ideal (--x  +
                8 1      2    4   1   1     2    3   2            8 1  
      -----------------------------------------------------------------------
                         3 3     95 2 2        3   3 2            2    
      10x x  + x x  + 1, -x x  + --x x  + 50x x  + -x x x  + 10x x x  +
         1 2    1 4      8 1 2    8 1 2      1 2   8 1 2 3      1 2 3  
      -----------------------------------------------------------------------
       2           2
      x x x  + 5x x x  + x x x x  + 1), {x , x })
       1 2 4     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,{- 4x  - 2x  + x , x , 2x  + x  + x , x }), ideal (- 3x  -
                    1     2    4   1    1    2    3   2               1  
      -----------------------------------------------------------------------
                            3       2 2       3     2           2       2    
      2x x  + x x  + 1, - 8x x  - 8x x  - 2x x  - 4x x x  - 2x x x  + 2x x x 
        1 2    1 4          1 2     1 2     1 2     1 2 3     1 2 3     1 2 4
      -----------------------------------------------------------------------
           2
      + x x x  + x x x x  + 1), {x , x })
         1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :