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     3             2                            11 2   3    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + 2x  + x , x }), ideal (--x  + -x x 
               9 1   2 2    4   1  3 1     2    3   2            9 1   2 1 2
     ------------------------------------------------------------------------
                  4 3     13 2 2       3   2 2       3   2     2 2      
     + x x  + 1, --x x  + --x x  + 3x x  + -x x x  + -x x x  + -x x x  +
        1 4      27 1 2    9 1 2     1 2   9 1 2 3   2 1 2 3   3 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

               7     1              7                     7              
o6 = (map(R,R,{-x  + -x  + x , x , --x  + 7x  + x , 3x  + -x  + x , x }),
               8 1   5 2    5   1  10 1     2    4    1   8 2    3   2   
     ------------------------------------------------------------------------
            7 2   1               3  343 3     147 2 2   147 2        21   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
     ------------------------------------------------------------------------
       21   2     21     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 })
       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} | 175000x_1x_2x_5^6-36750x_2^9x_5-56x_2^9+91875x_2^8x_5^
     {-9}  | 2240x_1x_2^2x_5^3-3675000x_1x_2x_5^5+11200x_1x_2x_5^4+
     {-9}  | 71680x_1x_2^3+117600000x_1x_2^2x_5^2+716800x_1x_2^2x_5
     {-3}  | 35x_1^2+8x_1x_2+40x_1x_5-40x_2^3                      
     ------------------------------------------------------------------------
                                                                           
     2+280x_2^8x_5-153125x_2^7x_5^3-1400x_2^7x_5^2+7000x_2^6x_5^3-35000x_2^
     771750x_2^9-1929375x_2^8x_5-1960x_2^8+3215625x_2^7x_5^2+19600x_2^7x_5-
     +42205078125000x_1x_2x_5^5-64312500000x_1x_2x_5^4+392000000x_1x_2x_5^3
                                                                           
     ------------------------------------------------------------------------
                                                                  
     5x_5^4+175000x_2^4x_5^5+40000x_2^2x_5^6+200000x_2x_5^7       
     147000x_2^6x_5^2+735000x_2^5x_5^3-3675000x_2^4x_5^4+11200x_2^
     +1792000x_1x_2x_5^2-8863066406250x_2^9+22157666015625x_2^8x_5
                                                                  
     ------------------------------------------------------------------------
                                                                     
                                                                     
     4x_5^3+512x_2^3x_5^3-840000x_2^2x_5^5+5120x_2^2x_5^4-4200000x_2x
     +33764062500x_2^8-36929443359375x_2^7x_5^2-281367187500x_2^7x_5+
                                                                     
     ------------------------------------------------------------------------
                                                                            
                                                                            
     _5^6+12800x_2x_5^5                                                     
     171500000x_2^7+1688203125000x_2^6x_5^2-2572500000x_2^6x_5-7840000x_2^6-
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     8441015625000x_2^5x_5^3+12862500000x_2^5x_5^2+39200000x_2^5x_5+358400x_2
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     ^5+42205078125000x_2^4x_5^4-64312500000x_2^4x_5^3+392000000x_2^4x_5^2+
                                                                           
     ------------------------------------------------------------------------
                                                                  
                                                                  
                                                                  
     1792000x_2^4x_5+16384x_2^4+26880000x_2^3x_5^2+245760x_2^3x_5+
                                                                  
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     9646875000000x_2^2x_5^5-14700000000x_2^2x_5^4+224000000x_2^2x_5^3+
                                                                       
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1228800x_2^2x_5^2+48234375000000x_2x_5^6-73500000000x_2x_5^5+448000000x_
                                                                             
     ------------------------------------------------------------------------
                            |
                            |
                            |
     2x_5^4+2048000x_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                   10                      11 2  
o13 = (map(R,R,{-x  + -x  + x , x , 8x  + --x  + x , x }), ideal (--x  +
                8 1   5 2    4   1    1    9 2    3   2            8 1  
      -----------------------------------------------------------------------
      7                   3     697 2 2   14   3   3 2       7   2    
      -x x  + x x  + 1, 3x x  + ---x x  + --x x  + -x x x  + -x x x  +
      5 1 2    1 4        1 2    60 1 2    9 1 2   8 1 2 3   5 1 2 3  
      -----------------------------------------------------------------------
        2       10   2
      8x x x  + --x x x  + x x x x  + 1), {x , x })
        1 2 4    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

                 3     1             9      7                      13 2  
o16 = (map(R,R,{--x  + -x  + x , x , -x  + --x  + x , x }), ideal (--x  +
                10 1   5 2    4   1  5 1   10 2    3   2           10 1  
      -----------------------------------------------------------------------
      1                 27 3      57 2 2    7   3    3 2       1   2    
      -x x  + x x  + 1, --x x  + ---x x  + --x x  + --x x x  + -x x x  +
      5 1 2    1 4      50 1 2   100 1 2   50 1 2   10 1 2 3   5 1 2 3  
      -----------------------------------------------------------------------
      9 2        7   2
      -x x x  + --x x x  + x x x x  + 1), {x , x })
      5 1 2 4   10 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  - x  + x , x , 4x  - 5x  + x , x }), ideal (5x  - x x  +
                  1    2    4   1    1     2    3   2             1    1 2  
      -----------------------------------------------------------------------
                   3        2 2       3     2          2       2      
      x x  + 1, 16x x  - 24x x  + 5x x  + 4x x x  - x x x  + 4x x x  -
       1 4         1 2      1 2     1 2     1 2 3    1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      5x 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 :