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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     3             7                            4 2   3      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + 9x  + x , x }), ideal (-x  + -x x  +
               3 1   5 2    4   1  4 1     2    3   2           3 1   5 1 2  
     ------------------------------------------------------------------------
                7 3     81 2 2   27   3   1 2       3   2     7 2      
     x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
      1 4      12 1 2   20 1 2    5 1 2   3 1 2 3   5 1 2 3   4 1 2 4  
     ------------------------------------------------------------------------
         2
     9x 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

               4     9                             1     3              
o6 = (map(R,R,{-x  + -x  + x , x , x  + 10x  + x , -x  + -x  + x , x }),
               5 1   5 2    5   1   1      2    4  5 1   8 2    3   2   
     ------------------------------------------------------------------------
            4 2   9               3   64 3     432 2 2   48 2       972   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x  +
            5 1   5 1 2    1 5    2  125 1 2   125 1 2   25 1 2 5   125 1 2  
     ------------------------------------------------------------------------
     216   2     12     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 })
      25 1 2 5    5 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} | 12500x_1x_2x_5^6-194400x_2^9x_5-236196x_2^9+54000x_2^8x_5^2+
     {-9}  | 43740x_1x_2^2x_5^3-10000x_1x_2x_5^5+24300x_1x_2x_5^4+155520x
     {-9}  | 13947137604x_1x_2^3+3188646000x_1x_2^2x_5^2+15496819560x_1x_
     {-3}  | 4x_1^2+9x_1x_2+5x_1x_5-5x_2^3                               
     ------------------------------------------------------------------------
                                                                            
     131220x_2^8x_5-10000x_2^7x_5^3-72900x_2^7x_5^2+40500x_2^6x_5^3-22500x_2
     _2^9-43200x_2^8x_5-34992x_2^8+8000x_2^7x_5^2+38880x_2^7x_5-32400x_2^6x_
     2^2x_5+200000000x_1x_2x_5^5-243000000x_1x_2x_5^4+1180980000x_1x_2x_5^3+
                                                                            
     ------------------------------------------------------------------------
                                                                         
     ^5x_5^4+12500x_2^4x_5^5+28125x_2^2x_5^6+15625x_2x_5^7               
     5^2+18000x_2^5x_5^3-10000x_2^4x_5^4+24300x_2^4x_5^3+98415x_2^3x_5^3-
     4304672100x_1x_2x_5^2-3110400000x_2^9+864000000x_2^8x_5+1049760000x_
                                                                         
     ------------------------------------------------------------------------
                                                                             
                                                                             
     22500x_2^2x_5^5+109350x_2^2x_5^4-12500x_2x_5^6+30375x_2x_5^5            
     2^8-160000000x_2^7x_5^2-972000000x_2^7x_5+472392000x_2^7+648000000x_2^6x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5^2-787320000x_2^6x_5-1913187600x_2^6-360000000x_2^5x_5^3+437400000x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5x_5^2+1062882000x_2^5x_5+7748409780x_2^5+200000000x_2^4x_5^4-243000000x
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     _2^4x_5^3+1180980000x_2^4x_5^2+4304672100x_2^4x_5+31381059609x_2^4+
                                                                        
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     7174453500x_2^3x_5^2+52301766015x_2^3x_5+450000000x_2^2x_5^5-546750000x_
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     2^2x_5^4+6643012500x_2^2x_5^3+29056536675x_2^2x_5^2+250000000x_2x_5^6-
                                                                           
     ------------------------------------------------------------------------
                                                             |
                                                             |
                                                             |
     303750000x_2x_5^5+1476225000x_2x_5^4+5380840125x_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

                                                                  2          
o13 = (map(R,R,{3x  + 4x  + x , x , x  + 2x  + x , x }), ideal (4x  + 4x x  +
                  1     2    4   1   1     2    3   2             1     1 2  
      -----------------------------------------------------------------------
                  3        2 2       3     2           2      2           2
      x x  + 1, 3x x  + 10x x  + 8x x  + 3x x x  + 4x x x  + x x x  + 2x 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

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     4             9                            7 2   4    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + 2x  + x , x }), ideal (-x  + -x x 
                2 1   3 2    4   1  5 1     2    3   2           2 1   3 1 2
      -----------------------------------------------------------------------
                  9 3     37 2 2   8   3   5 2       4   2     9 2      
      + x x  + 1, -x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      2 1 2    5 1 2   3 1 2   2 1 2 3   3 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

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