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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

               1                   1     10                      3 2        
o3 = (map(R,R,{-x  + 4x  + x , x , -x  + --x  + x , x }), ideal (-x  + 4x x 
               2 1     2    4   1  8 1    9 2    3   2           2 1     1 2
     ------------------------------------------------------------------------
                  1 3     19 2 2   40   3   1 2           2     1 2      
     + x x  + 1, --x x  + --x x  + --x x  + -x x x  + 4x x x  + -x x x  +
        1 4      16 1 2   18 1 2    9 1 2   2 1 2 3     1 2 3   8 1 2 4  
     ------------------------------------------------------------------------
     10   2
     --x x x  + x x x x  + 1), {x , x })
      9 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

               1     7             5     2              7                    
o6 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x  + -x  + x , x }), ideal
               2 1   8 2    5   1  2 1   3 2    4   1   9 2    3   2         
     ------------------------------------------------------------------------
      1 2   7               3  1 3     21 2 2   3 2       147   3   21   2  
     (-x  + -x x  + x x  - x , -x x  + --x x  + -x x x  + ---x x  + --x x x 
      2 1   8 1 2    1 5    2  8 1 2   32 1 2   4 1 2 5   128 1 2    8 1 2 5
     ------------------------------------------------------------------------
       3     2   343 4   147 3     21 2 2      3
     + -x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
       2 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_5^6-75264x_2^9x_5-16807x_2^9+
     {-9}  | 9604x_1x_2^2x_5^3-24576x_1x_2x_5^5+10976x_
     {-9}  | 7909306972x_1x_2^3+20239392768x_1x_2^2x_5^
     {-3}  | 4x_1^2+7x_1x_2+8x_1x_5-8x_2^3             
     ------------------------------------------------------------------------
                                                                           
     43008x_2^8x_5^2+19208x_2^8x_5-16384x_2^7x_5^3-21952x_2^7x_5^2+25088x_2
     1x_2x_5^4+56448x_2^9-32256x_2^8x_5-4802x_2^8+12288x_2^7x_5^2+10976x_2^
     2+18078415936x_1x_2^2x_5+77309411328x_1x_2x_5^5-17263755264x_1x_2x_5^4
                                                                           
     ------------------------------------------------------------------------
                                                                      
     ^6x_5^3-28672x_2^5x_5^4+32768x_2^4x_5^5+57344x_2^2x_5^6+65536x_2x
     7x_5-18816x_2^6x_5^2+21504x_2^5x_5^3-24576x_2^4x_5^4+10976x_2^4x_
     +15420489728x_1x_2x_5^3+10330523392x_1x_2x_5^2-177570054144x_2^9+
                                                                      
     ------------------------------------------------------------------------
                                                                             
     _5^7                                                                    
     5^3+16807x_2^3x_5^3-43008x_2^2x_5^5+38416x_2^2x_5^4-49152x_2x_5^6+21952x
     101468602368x_2^8x_5+22658678784x_2^8-38654705664x_2^7x_5^2-43159388160x
                                                                             
     ------------------------------------------------------------------------
                                                                       
                                                                       
     _2x_5^5                                                           
     _2^7x_5+3855122432x_2^7+59190018048x_2^6x_5^2-13217562624x_2^6x_5-
                                                                       
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5903156224x_2^6-67645734912x_2^5x_5^3+15105785856x_2^5x_5^2+6746464256x_
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     2^5x_5+9039207968x_2^5+77309411328x_2^4x_5^4-17263755264x_2^4x_5^3+
                                                                        
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     15420489728x_2^4x_5^2+10330523392x_2^4x_5+13841287201x_2^4+35418937344x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^3x_5^2+47455841832x_2^3x_5+135291469824x_2^2x_5^5-30211571712x_2^2x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     4+67464642560x_2^2x_5^3+54235247808x_2^2x_5^2+154618822656x_2x_5^6-
                                                                        
     ------------------------------------------------------------------------
                                                                 |
                                                                 |
                                                                 |
     34527510528x_2x_5^5+30840979456x_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

                6     7             1                           11 2   7    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , x }), ideal (--x  + -x x 
                5 1   6 2    4   1  3 1    2    3   2            5 1   6 1 2
      -----------------------------------------------------------------------
                  2 3     143 2 2   7   3   6 2       7   2     1 2      
      + x x  + 1, -x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      5 1 2    90 1 2   6 1 2   5 1 2 3   6 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
         2
      x x x  + x x x x  + 1), {x , x })
       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             1     4                      11 2   1    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                8 1   5 2    4   1  9 1   9 2    3   2            8 1   5 1 2
      -----------------------------------------------------------------------
                   1 3     17 2 2    4   3   3 2       1   2     1 2      
      + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      24 1 2   90 1 2   45 1 2   8 1 2 3   5 1 2 3   9 1 2 4  
      -----------------------------------------------------------------------
      4   2
      -x x x  + x x x x  + 1), {x , x })
      9 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

                                                                  2         
o19 = (map(R,R,{2x  + x  + x , x , - x  + x  + x , x }), ideal (3x  + x x  +
                  1    2    4   1     1    2    3   2             1    1 2  
      -----------------------------------------------------------------------
                    3      2 2      3     2          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    1 2    1 2     1 2 3    1 2 3    1 2 4    1 2 4  
      -----------------------------------------------------------------------
      x x x x  + 1), {x , x })
       1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :