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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

               4                         1                      9 2          
o3 = (map(R,R,{-x  + 2x  + x , x , 4x  + -x  + x , x }), ideal (-x  + 2x x  +
               5 1     2    4   1    1   4 2    3   2           5 1     1 2  
     ------------------------------------------------------------------------
               16 3     41 2 2   1   3   4 2           2       2      
     x x  + 1, --x x  + --x x  + -x x  + -x x x  + 2x x x  + 4x x x  +
      1 4       5 1 2    5 1 2   2 1 2   5 1 2 3     1 2 3     1 2 4  
     ------------------------------------------------------------------------
     1   2
     -x x x  + x x x x  + 1), {x , x })
     4 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              9               7     9              
o6 = (map(R,R,{-x  + -x  + x , x , --x  + x  + x , --x  + -x  + x , x }),
               3 1   9 2    5   1  10 1    2    4  10 1   2 2    3   2   
     ------------------------------------------------------------------------
            1 2   7               3   1 3      7 2 2   1 2       49   3  
     ideal (-x  + -x x  + x x  - x , --x x  + --x x  + -x x x  + --x x  +
            3 1   9 1 2    1 5    2  27 1 2   27 1 2   3 1 2 5   81 1 2  
     ------------------------------------------------------------------------
     14   2          2   343 4   49 3     7 2 2      3
     --x x x  + x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
      9 1 2 5    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} | 59049x_1x_2x_5^6-71442x_2^9x_5-16807x_2^9+45927x_2^8x_5^2
     {-9}  | 7203x_1x_2^2x_5^3-19683x_1x_2x_5^5+9261x_1x_2x_5^4+23814x
     {-9}  | 5931980229x_1x_2^3+16209796869x_1x_2^2x_5^2+15253663446x_
     {-3}  | 3x_1^2+7x_1x_2+9x_1x_5-9x_2^3                            
     ------------------------------------------------------------------------
                                                                             
     +21609x_2^8x_5-19683x_2^7x_5^3-27783x_2^7x_5^2+35721x_2^6x_5^3-45927x_2^
     _2^9-15309x_2^8x_5-2401x_2^8+6561x_2^7x_5^2+6174x_2^7x_5-11907x_2^6x_5^2
     1x_2^2x_5+62762119218x_1x_2x_5^5-14765025303x_1x_2x_5^4+13894111602x_1x_
                                                                             
     ------------------------------------------------------------------------
                                                                     
     5x_5^4+59049x_2^4x_5^5+137781x_2^2x_5^6+177147x_2x_5^7          
     +15309x_2^5x_5^3-19683x_2^4x_5^4+9261x_2^4x_5^3+16807x_2^3x_5^3-
     2x_5^3+9805926501x_1x_2x_5^2-75934415844x_2^9+48814981614x_2^8x_
                                                                     
     ------------------------------------------------------------------------
                                                                             
                                                                             
     45927x_2^2x_5^5+43218x_2^2x_5^4-59049x_2x_5^6+27783x_2x_5^5             
     5+11483908569x_2^8-20920706406x_2^7x_5^2-24608375505x_2^7x_5+2315685267x
                                                                             
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     _2^7+37967207922x_2^6x_5^2-8931928887x_2^6x_5-4202539929x_2^6-
                                                                   
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     48814981614x_2^5x_5^3+11483908569x_2^5x_5^2+5403265623x_2^5x_5+
                                                                    
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     7626831723x_2^5+62762119218x_2^4x_5^4-14765025303x_2^4x_5^3+13894111602x
                                                                             
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     _2^4x_5^2+9805926501x_2^4x_5+13841287201x_2^4+37822859361x_2^3x_5^2+
                                                                         
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     53387822061x_2^3x_5+146444944842x_2^2x_5^5-34451725707x_2^2x_5^4+
                                                                      
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     81048984345x_2^2x_5^3+68641485507x_2^2x_5^2+188286357654x_2x_5^6-
                                                                      
     ------------------------------------------------------------------------
                                                                 |
                                                                 |
                                                                 |
     44295075909x_2x_5^5+41682334806x_2x_5^4+29417779503x_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

                1                        3                      3 2          
o13 = (map(R,R,{-x  + 9x  + x , x , x  + -x  + x , x }), ideal (-x  + 9x x  +
                2 1     2    4   1   1   2 2    3   2           2 1     1 2  
      -----------------------------------------------------------------------
                1 3     39 2 2   27   3   1 2           2      2      
      x x  + 1, -x x  + --x x  + --x x  + -x x x  + 9x x x  + x x x  +
       1 4      2 1 2    4 1 2    2 1 2   2 1 2 3     1 2 3    1 2 4  
      -----------------------------------------------------------------------
      3   2
      -x x x  + x x x x  + 1), {x , x })
      2 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

                      2             1     4                        2   2    
o16 = (map(R,R,{2x  + -x  + x , x , -x  + -x  + x , x }), ideal (3x  + -x x 
                  1   3 2    4   1  3 1   3 2    3   2             1   3 1 2
      -----------------------------------------------------------------------
                  2 3     26 2 2   8   3     2       2   2     1 2      
      + x x  + 1, -x x  + --x x  + -x x  + 2x x x  + -x x x  + -x x x  +
         1 4      3 1 2    9 1 2   9 1 2     1 2 3   3 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      4   2
      -x x x  + x x x x  + 1), {x , x })
      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

                                                                       2  
o19 = (map(R,R,{- 4x  - 4x  + x , x , 6x  + 4x  + x , x }), ideal (- 3x  -
                    1     2    4   1    1     2    3   2               1  
      -----------------------------------------------------------------------
                             3        2 2        3     2           2    
      4x x  + x x  + 1, - 24x x  - 40x x  - 16x x  - 4x x x  - 4x x x  +
        1 2    1 4           1 2      1 2      1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      6x x x  + 4x 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 :