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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     1             3     4                      14 2   1    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
               9 1   3 2    4   1  2 1   7 2    3   2            9 1   3 1 2
     ------------------------------------------------------------------------
                 5 3     103 2 2    4   3   5 2       1   2     3 2      
     + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
        1 4      6 1 2   126 1 2   21 1 2   9 1 2 3   3 1 2 3   2 1 2 4  
     ------------------------------------------------------------------------
     4   2
     -x x x  + x x x x  + 1), {x , x })
     7 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

               8     10             1     4         1     8              
o6 = (map(R,R,{-x  + --x  + x , x , -x  + -x  + x , -x  + -x  + x , x }),
               9 1    9 2    5   1  2 1   9 2    4  2 1   9 2    3   2   
     ------------------------------------------------------------------------
            8 2   10               3  512 3     640 2 2   64 2       800   3
     ideal (-x  + --x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x 
            9 1    9 1 2    1 5    2  729 1 2   243 1 2   27 1 2 5   243 1 2
     ------------------------------------------------------------------------
       160   2     8     2   1000 4   100 3     10 2 2      3
     + ---x x x  + -x x x  + ----x  + ---x x  + --x x  + x x ), {x , x , x })
        27 1 2 5   3 1 2 5    729 2    27 2 5    3 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 472392x_1x_2x_5^6-3110400x_2^9x_5-800000x_2^9+1399680x_2^8x_5^2+
     {-9}  | 90000x_1x_2^2x_5^3-157464x_1x_2x_5^5+81000x_1x_2x_5^4+1036800x_2
     {-9}  | 1562500000x_1x_2^3+2733750000x_1x_2^2x_5^2+2812500000x_1x_2^2x_5
     {-3}  | 8x_1^2+10x_1x_2+9x_1x_5-9x_2^3                                  
     ------------------------------------------------------------------------
                                                                
     720000x_2^8x_5-419904x_2^7x_5^3-648000x_2^7x_5^2+583200x_2^
     ^9-466560x_2^8x_5-80000x_2^8+139968x_2^7x_5^2+144000x_2^7x_
     +6198727824x_1x_2x_5^5-1594323000x_1x_2x_5^4+1640250000x_1x
                                                                
     ------------------------------------------------------------------------
                                                                       
     6x_5^3-524880x_2^5x_5^4+472392x_2^4x_5^5+590490x_2^2x_5^6+531441x_
     5-194400x_2^6x_5^2+174960x_2^5x_5^3-157464x_2^4x_5^4+81000x_2^4x_5
     _2x_5^3+1265625000x_1x_2x_5^2-40814668800x_2^9+18366600960x_2^8x_5
                                                                       
     ------------------------------------------------------------------------
                                                                          
     2x_5^7                                                               
     ^3+112500x_2^3x_5^3-196830x_2^2x_5^5+202500x_2^2x_5^4-177147x_2x_5^6+
     +4723920000x_2^8-5509980288x_2^7x_5^2-7085880000x_2^7x_5+729000000x_2
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
     91125x_2x_5^5                                                           
     ^7+7652750400x_2^6x_5^2-1968300000x_2^6x_5-1012500000x_2^6-6887475360x_2
                                                                             
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     ^5x_5^3+1771470000x_2^5x_5^2+911250000x_2^5x_5+1406250000x_2^5+
                                                                    
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     6198727824x_2^4x_5^4-1594323000x_2^4x_5^3+1640250000x_2^4x_5^2+
                                                                    
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1265625000x_2^4x_5+1953125000x_2^4+3417187500x_2^3x_5^2+5273437500x_2^3x
                                                                             
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     _5+7748409780x_2^2x_5^5-1992903750x_2^2x_5^4+5125781250x_2^2x_5^3+
                                                                       
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     4746093750x_2^2x_5^2+6973568802x_2x_5^6-1793613375x_2x_5^5+1845281250x_
                                                                            
     ------------------------------------------------------------------------
                               |
                               |
                               |
     2x_5^4+1423828125x_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

                      1             5     2                        2   1    
o13 = (map(R,R,{2x  + -x  + x , x , -x  + -x  + x , x }), ideal (3x  + -x x 
                  1   6 2    4   1  2 1   3 2    3   2             1   6 1 2
      -----------------------------------------------------------------------
                    3     7 2 2   1   3     2       1   2     5 2      
      + x x  + 1, 5x x  + -x x  + -x x  + 2x x x  + -x x x  + -x x x  +
         1 4        1 2   4 1 2   9 1 2     1 2 3   6 1 2 3   2 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

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      9             1     4                      7 2  
o16 = (map(R,R,{-x  + --x  + x , x , -x  + -x  + x , x }), ideal (-x  +
                2 1   10 2    4   1  9 1   3 2    3   2           2 1  
      -----------------------------------------------------------------------
       9                  5 3     103 2 2   6   3   5 2        9   2    
      --x x  + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + --x x x  +
      10 1 2    1 4      18 1 2    30 1 2   5 1 2   2 1 2 3   10 1 2 3  
      -----------------------------------------------------------------------
      1 2       4   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      9 1 2 4   3 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
--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: 4
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                   2         
o19 = (map(R,R,{4x  - x  + x , x , - 4x  - x  + x , x }), ideal (5x  - x x  +
                  1    2    4   1      1    2    3   2             1    1 2  
      -----------------------------------------------------------------------
                     3        3     2          2       2          2
      x x  + 1, - 16x x  + x x  + 4x x x  - x x x  - 4x x x  - x x x  +
       1 4           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 :