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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

               7     7                  3                      15 2   7      
o3 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (--x  + -x x  +
               8 1   3 2    4   1   1   4 2    3   2            8 1   3 1 2  
     ------------------------------------------------------------------------
               7 3     287 2 2   7   3   7 2       7   2      2       3   2
     x x  + 1, -x x  + ---x x  + -x x  + -x x x  + -x x x  + x x x  + -x x x 
      1 4      8 1 2    96 1 2   4 1 2   8 1 2 3   3 1 2 3    1 2 4   4 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

               1                        1          1     5              
o6 = (map(R,R,{-x  + 3x  + x , x , x  + -x  + x , --x  + -x  + x , x }),
               5 1     2    5   1   1   5 2    4  10 1   9 2    3   2   
     ------------------------------------------------------------------------
            1 2                   3   1  3      9 2 2    3 2       27   3  
     ideal (-x  + 3x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
            5 1     1 2    1 5    2  125 1 2   25 1 2   25 1 2 5    5 1 2  
     ------------------------------------------------------------------------
     18   2     3     2      4      3       2 2      3
     --x x x  + -x x x  + 27x  + 27x x  + 9x x  + x x ), {x , x , x })
      5 1 2 5   5 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} | 5x_1x_2x_5^6-54x_2^9x_5-1215x_2^9+9x_2^8x_5^2+405x_2^8x_5-x_2^
     {-9}  | 675x_1x_2^2x_5^3-5x_1x_2x_5^5+225x_1x_2x_5^4+54x_2^9-9x_2^8x_5
     {-9}  | 12301875x_1x_2^3+91125x_1x_2^2x_5^2+8201250x_1x_2^2x_5+10x_1x_
     {-3}  | x_1^2+15x_1x_2+5x_1x_5-5x_2^3                                 
     ------------------------------------------------------------------------
                                                                           
     7x_5^3-135x_2^7x_5^2+45x_2^6x_5^3-15x_2^5x_5^4+5x_2^4x_5^5+75x_2^2x_5^
     -135x_2^8+x_2^7x_5^2+90x_2^7x_5-45x_2^6x_5^2+15x_2^5x_5^3-5x_2^4x_5^4+
     2x_5^5-225x_1x_2x_5^4+20250x_1x_2x_5^3+1366875x_1x_2x_5^2-108x_2^9+18x
                                                                           
     ------------------------------------------------------------------------
                                                                          
     6+25x_2x_5^7                                                         
     225x_2^4x_5^3+10125x_2^3x_5^3-75x_2^2x_5^5+6750x_2^2x_5^4-25x_2x_5^6+
     _2^8x_5+405x_2^8-2x_2^7x_5^2-225x_2^7x_5+2025x_2^7+90x_2^6x_5^2-2025x
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
     1125x_2x_5^5                                                            
     _2^6x_5-91125x_2^6-30x_2^5x_5^3+675x_2^5x_5^2+30375x_2^5x_5+4100625x_2^5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     +10x_2^4x_5^4-225x_2^4x_5^3+20250x_2^4x_5^2+1366875x_2^4x_5+184528125x_2
                                                                             
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     ^4+1366875x_2^3x_5^2+184528125x_2^3x_5+150x_2^2x_5^5-3375x_2^2x_5^4+
                                                                         
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     759375x_2^2x_5^3+61509375x_2^2x_5^2+50x_2x_5^6-1125x_2x_5^5+101250x_2x_5
                                                                             
     ------------------------------------------------------------------------
                        |
                        |
                        |
     ^4+6834375x_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

                3     3             2                            5 2   3    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + 2x  + x , x }), ideal (-x  + -x x 
                2 1   5 2    4   1  5 1     2    3   2           2 1   5 1 2
      -----------------------------------------------------------------------
                  3 3     81 2 2   6   3   3 2       3   2     2 2      
      + x x  + 1, -x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      5 1 2   25 1 2   5 1 2   2 1 2 3   5 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

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

                      1             9     1                        2   1    
o16 = (map(R,R,{2x  + -x  + x , x , -x  + -x  + x , x }), ideal (3x  + -x x 
                  1   5 2    4   1  2 1   7 2    3   2             1   5 1 2
      -----------------------------------------------------------------------
                    3     83 2 2    1   3     2       1   2     9 2      
      + x x  + 1, 9x x  + --x x  + --x x  + 2x x x  + -x x x  + -x x x  +
         1 4        1 2   70 1 2   35 1 2     1 2 3   5 1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      1   2
      -x x x  + x x x x  + 1), {x , x })
      7 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  + 2x  + x , x , 3x  - 2x  + x , x }), ideal (- 5x  +
                    1     2    4   1    1     2    3   2               1  
      -----------------------------------------------------------------------
                             3        2 2       3     2           2    
      2x x  + x x  + 1, - 18x x  + 18x x  - 4x x  - 6x x x  + 2x x x  +
        1 2    1 4           1 2      1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      3x x x  - 2x x x  + x x x x  + 1), {x , x })
        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 :