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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

               5     5                   2                      9 2   5      
o3 = (map(R,R,{-x  + -x  + x , x , 2x  + -x  + x , x }), ideal (-x  + -x x  +
               4 1   8 2    4   1    1   3 2    3   2           4 1   8 1 2  
     ------------------------------------------------------------------------
               5 3     25 2 2    5   3   5 2       5   2       2      
     x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + 2x x x  +
      1 4      2 1 2   12 1 2   12 1 2   4 1 2 3   8 1 2 3     1 2 4  
     ------------------------------------------------------------------------
     2   2
     -x x x  + x x x x  + 1), {x , x })
     3 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

               5     9             2              1                          
o6 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , -x  + 5x  + x , x }), ideal
               2 1   4 2    5   1  3 1    2    4  3 1     2    3   2         
     ------------------------------------------------------------------------
      5 2   9               3  125 3     675 2 2   75 2       1215   3  
     (-x  + -x x  + x x  - x , ---x x  + ---x x  + --x x x  + ----x x  +
      2 1   4 1 2    1 5    2   8  1 2    16 1 2    4 1 2 5    32  1 2  
     ------------------------------------------------------------------------
     135   2     15     2   729 4   243 3     27 2 2      3
     ---x x x  + --x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
      4  1 2 5    2 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} | 5120x_1x_2x_5^6-388800x_2^9x_5-295245x_2^9+86400x_2^8x_5^2
     {-9}  | 21870x_1x_2^2x_5^3-6400x_1x_2x_5^5+9720x_1x_2x_5^4+486000x
     {-9}  | 34867844010x_1x_2^3+10203667200x_1x_2^2x_5^2+30993639120x_
     {-3}  | 10x_1^2+9x_1x_2+4x_1x_5-4x_2^3                            
     ------------------------------------------------------------------------
                                                                             
     +131220x_2^8x_5-12800x_2^7x_5^3-58320x_2^7x_5^2+25920x_2^6x_5^3-11520x_2
     _2^9-108000x_2^8x_5-54675x_2^8+16000x_2^7x_5^2+48600x_2^7x_5-32400x_2^6x
     1x_2^2x_5+1310720000x_1x_2x_5^5-995328000x_1x_2x_5^4+3023308800x_1x_2x_5
                                                                             
     ------------------------------------------------------------------------
                                                                   
     ^5x_5^4+5120x_2^4x_5^5+4608x_2^2x_5^6+2048x_2x_5^7            
     _5^2+14400x_2^5x_5^3-6400x_2^4x_5^4+9720x_2^4x_5^3+19683x_2^3x
     ^3+6887475360x_1x_2x_5^2-99532800000x_2^9+22118400000x_2^8x_5+
                                                                   
     ------------------------------------------------------------------------
                                                                             
                                                                             
     _5^3-5760x_2^2x_5^5+17496x_2^2x_5^4-2560x_2x_5^6+3888x_2x_5^5           
     16796160000x_2^8-3276800000x_2^7x_5^2-12441600000x_2^7x_5+3779136000x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     7+6635520000x_2^6x_5^2-5038848000x_2^6x_5-7652750400x_2^6-2949120000x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     5x_5^3+2239488000x_2^5x_5^2+3401222400x_2^5x_5+15496819560x_2^5+
                                                                     
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     1310720000x_2^4x_5^4-995328000x_2^4x_5^3+3023308800x_2^4x_5^2+
                                                                   
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     6887475360x_2^4x_5+31381059609x_2^4+9183300480x_2^3x_5^2+41841412812x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     3x_5+1179648000x_2^2x_5^5-895795200x_2^2x_5^4+6802444800x_2^2x_5^3+
                                                                        
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     18596183472x_2^2x_5^2+524288000x_2x_5^6-398131200x_2x_5^5+1209323520x_2x
                                                                             
     ------------------------------------------------------------------------
                             |
                             |
                             |
     _5^4+2754990144x_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

                5     4             1     9                      9 2   4    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                4 1   9 2    4   1  5 1   2 2    3   2           4 1   9 1 2
      -----------------------------------------------------------------------
                  1 3     2057 2 2       3   5 2       4   2     1 2      
      + x x  + 1, -x x  + ----x x  + 2x x  + -x x x  + -x x x  + -x x x  +
         1 4      4 1 2    360 1 2     1 2   4 1 2 3   9 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      9   2
      -x x x  + x x x x  + 1), {x , x })
      2 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             1                            16 2   9    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + 2x  + x , x }), ideal (--x  + -x x 
                9 1   8 2    4   1  5 1     2    3   2            9 1   8 1 2
      -----------------------------------------------------------------------
                   7 3     641 2 2   9   3   7 2       9   2     1 2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      45 1 2   360 1 2   4 1 2   9 1 2 3   8 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

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,{- 6x  - 3x  + x , x , - x  + x , x }), ideal (- 5x  - 3x x  +
                    1     2    4   1     2    3   2               1     1 2  
      -----------------------------------------------------------------------
                  2 2       3     2           2        2
      x x  + 1, 6x x  + 3x x  - 6x x x  - 3x x x  - x x x  + x x x x  + 1),
       1 4        1 2     1 2     1 2 3     1 2 3    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 :