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

               2     7             8      1                      11 2   7    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (--x  + -x x 
               9 1   3 2    4   1  3 1   10 2    3   2            9 1   3 1 2
     ------------------------------------------------------------------------
                 16 3     281 2 2    7   3   2 2       7   2     8 2      
     + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
        1 4      27 1 2    45 1 2   30 1 2   9 1 2 3   3 1 2 3   3 1 2 4  
     ------------------------------------------------------------------------
      1   2
     --x x x  + x x x x  + 1), {x , x })
     10 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

               8     10             1               1     8              
o6 = (map(R,R,{-x  + --x  + x , x , -x  + 6x  + x , -x  + -x  + x , x }),
               9 1    3 2    5   1  2 1     2    4  5 1   3 2    3   2   
     ------------------------------------------------------------------------
            8 2   10               3  512 3     640 2 2   64 2       800   3
     ideal (-x  + --x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x 
            9 1    3 1 2    1 5    2  729 1 2    81 1 2   27 1 2 5    27 1 2
     ------------------------------------------------------------------------
       160   2     8     2   1000 4   100 3        2 2      3
     + ---x x x  + -x x x  + ----x  + ---x x  + 10x x  + x x ), {x , x , x })
        9  1 2 5   3 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} | 1944x_1x_2x_5^6-115200x_2^9x_5-800000x_2^9+17280x_2^8x_5^2+
     {-9}  | 90000x_1x_2^2x_5^3-1944x_1x_2x_5^5+27000x_1x_2x_5^4+115200x
     {-9}  | 1562500000x_1x_2^3+33750000x_1x_2^2x_5^2+937500000x_1x_2^2x
     {-3}  | 8x_1^2+30x_1x_2+9x_1x_5-9x_2^3                             
     ------------------------------------------------------------------------
                                                                           
     240000x_2^8x_5-1728x_2^7x_5^3-72000x_2^7x_5^2+21600x_2^6x_5^3-6480x_2^
     _2^9-17280x_2^8x_5-80000x_2^8+1728x_2^7x_5^2+48000x_2^7x_5-21600x_2^6x
     _5+34992x_1x_2x_5^5-243000x_1x_2x_5^4+6750000x_1x_2x_5^3+140625000x_1x
                                                                           
     ------------------------------------------------------------------------
                                                                      
     5x_5^4+1944x_2^4x_5^5+7290x_2^2x_5^6+2187x_2x_5^7                
     _5^2+6480x_2^5x_5^3-1944x_2^4x_5^4+27000x_2^4x_5^3+337500x_2^3x_5
     _2x_5^2-2073600x_2^9+311040x_2^8x_5+2160000x_2^8-31104x_2^7x_5^2-
                                                                      
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ^3-7290x_2^2x_5^5+202500x_2^2x_5^4-2187x_2x_5^6+30375x_2x_5^5           
     1080000x_2^7x_5+3000000x_2^7+388800x_2^6x_5^2-2700000x_2^6x_5-37500000x_
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     2^6-116640x_2^5x_5^3+810000x_2^5x_5^2+11250000x_2^5x_5+468750000x_2^5+
                                                                           
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     34992x_2^4x_5^4-243000x_2^4x_5^3+6750000x_2^4x_5^2+140625000x_2^4x_5+
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5859375000x_2^4+126562500x_2^3x_5^2+5273437500x_2^3x_5+131220x_2^2x_5^5-
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     911250x_2^2x_5^4+63281250x_2^2x_5^3+1582031250x_2^2x_5^2+39366x_2x_5^6-
                                                                            
     ------------------------------------------------------------------------
                                                      |
                                                      |
                                                      |
     273375x_2x_5^5+7593750x_2x_5^4+158203125x_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

                     7             3      7                        2   7    
o13 = (map(R,R,{x  + -x  + x , x , -x  + --x  + x , x }), ideal (2x  + -x x 
                 1   3 2    4   1  8 1   10 2    3   2             1   3 1 2
      -----------------------------------------------------------------------
                  3 3     63 2 2   49   3    2       7   2     3 2      
      + x x  + 1, -x x  + --x x  + --x x  + x x x  + -x x x  + -x x x  +
         1 4      8 1 2   40 1 2   30 1 2    1 2 3   3 1 2 3   8 1 2 4  
      -----------------------------------------------------------------------
       7   2
      --x x x  + x x x x  + 1), {x , x })
      10 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

                5     5             3                           7 2   5      
o16 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , x }), ideal (-x  + -x x  +
                2 1   2 2    4   1  4 1    2    3   2           2 1   2 1 2  
      -----------------------------------------------------------------------
                15 3     35 2 2   5   3   5 2       5   2     3 2      
      x x  + 1, --x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  +
       1 4       8 1 2    8 1 2   2 1 2   2 1 2 3   2 1 2 3   4 1 2 4  
      -----------------------------------------------------------------------
         2
      x 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
--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
--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: 4
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                      2  
o19 = (map(R,R,{- 8x  - 5x  + x , x , 8x  + x  + x , x }), ideal (- 7x  -
                    1     2    4   1    1    2    3   2               1  
      -----------------------------------------------------------------------
                             3        2 2       3     2           2    
      5x x  + x x  + 1, - 64x x  - 48x x  - 5x x  - 8x x x  - 5x x x  +
        1 2    1 4           1 2      1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2          2
      8x x x  + x 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 :