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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     5                        2   1      
o3 = (map(R,R,{2x  + -x  + x , x , -x  + -x  + x , x }), ideal (3x  + -x x  +
                 1   5 2    4   1  2 1   2 2    3   2             1   5 1 2  
     ------------------------------------------------------------------------
                3     51 2 2   1   3     2       1   2     1 2       5   2
     x x  + 1, x x  + --x x  + -x x  + 2x x x  + -x x x  + -x x x  + -x x x 
      1 4       1 2   10 1 2   2 1 2     1 2 3   5 1 2 3   2 1 2 4   2 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

               9                                        8                    
o6 = (map(R,R,{-x  + 2x  + x , x , x  + 6x  + x , 2x  + -x  + x , x }), ideal
               7 1     2    5   1   1     2    4    1   9 2    3   2         
     ------------------------------------------------------------------------
      9 2                   3  729 3     486 2 2   243 2       108   3  
     (-x  + 2x x  + x x  - x , ---x x  + ---x x  + ---x x x  + ---x x  +
      7 1     1 2    1 5    2  343 1 2    49 1 2    49 1 2 5    7  1 2  
     ------------------------------------------------------------------------
     108   2     27     2     4      3       2 2      3
     ---x x x  + --x x x  + 8x  + 12x x  + 6x x  + x x ), {x , x , x })
      7  1 2 5    7 1 2 5     2      2 5     2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                    
     {-10} | 63x_1x_2x_5^6-1944x_2^9x_5-2016x_2^9+486x_2^8x_5^2+1008x_2
     {-9}  | 7056x_1x_2^2x_5^3-1701x_1x_2x_5^5+3528x_1x_2x_5^4+52488x_2
     {-9}  | 22127616x_1x_2^3+5334336x_1x_2^2x_5^2+22127616x_1x_2^2x_5+
     {-3}  | 9x_1^2+14x_1x_2+7x_1x_5-7x_2^3                            
     ------------------------------------------------------------------------
                                                                           
     ^8x_5-81x_2^7x_5^3-504x_2^7x_5^2+252x_2^6x_5^3-126x_2^5x_5^4+63x_2^4x_
     ^9-13122x_2^8x_5-9072x_2^8+2187x_2^7x_5^2+9072x_2^7x_5-6804x_2^6x_5^2+
     413343x_1x_2x_5^5-428652x_1x_2x_5^4+1778112x_1x_2x_5^3+5531904x_1x_2x_
                                                                           
     ------------------------------------------------------------------------
                                                                             
     5^5+98x_2^2x_5^6+49x_2x_5^7                                             
     3402x_2^5x_5^3-1701x_2^4x_5^4+3528x_2^4x_5^3+10976x_2^3x_5^3-2646x_2^2x_
     5^2-12754584x_2^9+3188646x_2^8x_5+3306744x_2^8-531441x_2^7x_5^2-2755620x
                                                                             
     ------------------------------------------------------------------------
                                                                         
                                                                         
     5^5+10976x_2^2x_5^4-1323x_2x_5^6+2744x_2x_5^5                       
     _2^7x_5+1143072x_2^7+1653372x_2^6x_5^2-1714608x_2^6x_5-3556224x_2^6-
                                                                         
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     826686x_2^5x_5^3+857304x_2^5x_5^2+1778112x_2^5x_5+11063808x_2^5+413343x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^4x_5^4-428652x_2^4x_5^3+1778112x_2^4x_5^2+5531904x_2^4x_5+34420736x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     4+8297856x_2^3x_5^2+51631104x_2^3x_5+642978x_2^2x_5^5-666792x_2^2x_5^4+
                                                                            
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     6914880x_2^2x_5^3+25815552x_2^2x_5^2+321489x_2x_5^6-333396x_2x_5^5+
                                                                        
     ------------------------------------------------------------------------
                                     |
                                     |
                                     |
     1382976x_2x_5^4+4302592x_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     7             2      9                      8 2   7    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (-x  + -x x 
                3 1   9 2    4   1  7 1   10 2    3   2           3 1   9 1 2
      -----------------------------------------------------------------------
                  10 3     31 2 2    7   3   5 2       7   2     2 2      
      + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      21 1 2   18 1 2   10 1 2   3 1 2 3   9 1 2 3   7 1 2 4  
      -----------------------------------------------------------------------
       9   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

                9     5             8     1                      14 2   5    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                5 1   4 2    4   1  7 1   2 2    3   2            5 1   4 1 2
      -----------------------------------------------------------------------
                  72 3     163 2 2   5   3   9 2       5   2     8 2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      35 1 2    70 1 2   8 1 2   5 1 2 3   4 1 2 3   7 1 2 4  
      -----------------------------------------------------------------------
      1   2
      -x x x  + x x x x  + 1), {x , x })
      2 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 , - x  + x , x }), ideal (5x  - 2x x  +
                  1     2    4   1     2    3   2             1     1 2  
      -----------------------------------------------------------------------
                    2 2       3     2           2        2
      x x  + 1, - 4x x  + 2x x  + 4x x x  - 2x 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 :