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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     3             1     1                      6 2   3      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x  +
               5 1   7 2    4   1  5 1   2 2    3   2           5 1   7 1 2  
     ------------------------------------------------------------------------
                1 3     13 2 2    3   3   1 2       3   2     1 2      
     x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
      1 4      25 1 2   70 1 2   14 1 2   5 1 2 3   7 1 2 3   5 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

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         1                    
o6 = (map(R,R,{5x  + -x  + x , x , -x  + -x  + x , -x  + 2x  + x , x }),
                 1   9 2    5   1  9 1   2 2    4  5 1     2    3   2   
     ------------------------------------------------------------------------
              2   1               3      3     25 2 2      2        5   3  
     ideal (5x  + -x x  + x x  - x , 125x x  + --x x  + 75x x x  + --x x  +
              1   9 1 2    1 5    2      1 2    3 1 2      1 2 5   27 1 2  
     ------------------------------------------------------------------------
     10   2            2    1  4    1 3     1 2 2      3
     --x x x  + 15x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
      3 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} | 295245x_1x_2x_5^6-109350x_2^9x_5-5x_2^9+492075x
     {-9}  | 45x_1x_2^2x_5^3-4428675x_1x_2x_5^5+405x_1x_2x_5
     {-9}  | 45x_1x_2^3+4428675x_1x_2^2x_5^2+810x_1x_2^2x_5+
     {-3}  | 45x_1^2+x_1x_2+9x_1x_5-9x_2^3                  
     ------------------------------------------------------------------------
                                                                         
     _2^8x_5^2+45x_2^8x_5-1476225x_2^7x_5^3-405x_2^7x_5^2+3645x_2^6x_5^3-
     ^4+1640250x_2^9-7381125x_2^8x_5-225x_2^8+22143375x_2^7x_5^2+4050x_2^
     3177332285411250x_1x_2x_5^5-145282683375x_1x_2x_5^4+26572050x_1x_2x_
                                                                         
     ------------------------------------------------------------------------
                                                                        
     32805x_2^5x_5^4+295245x_2^4x_5^5+6561x_2^2x_5^6+59049x_2x_5^7      
     7x_5-54675x_2^6x_5^2+492075x_2^5x_5^3-4428675x_2^4x_5^4+405x_2^4x_5
     5^3+3645x_1x_2x_5^2-1176789735337500x_2^9+5295553809018750x_2^8x_5+
                                                                        
     ------------------------------------------------------------------------
                                                                         
                                                                         
     ^3+x_2^3x_5^3-98415x_2^2x_5^5+18x_2^2x_5^4-885735x_2x_5^6+81x_2x_5^5
     242137805625x_2^8-15886661427056250x_2^7x_5^2-3632067084375x_2^7x_5+
                                                                         
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     66430125x_2^7+39226324511250x_2^6x_5^2-1793613375x_2^6x_5-164025x_2^6-
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     353036920601250x_2^5x_5^3+16142520375x_2^5x_5^2+1476225x_2^5x_5+405x_2^5
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     +3177332285411250x_2^4x_5^4-145282683375x_2^4x_5^3+26572050x_2^4x_5^2+
                                                                           
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     3645x_2^4x_5+x_2^4+98415x_2^3x_5^2+27x_2^3x_5+70607384120250x_2^2x_5^5-
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3228504075x_2^2x_5^4+1476225x_2^2x_5^3+243x_2^2x_5^2+635466457082250x_2x
                                                                             
     ------------------------------------------------------------------------
                                                          |
                                                          |
                                                          |
     _5^6-29056536675x_2x_5^5+5314410x_2x_5^4+729x_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                    7     1                      3 2        
o13 = (map(R,R,{-x  + 9x  + x , x , --x  + -x  + x , x }), ideal (-x  + 9x x 
                2 1     2    4   1  10 1   4 2    3   2           2 1     1 2
      -----------------------------------------------------------------------
                   7 3     257 2 2   9   3   1 2           2      7 2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + 9x x x  + --x x x  +
         1 4      20 1 2    40 1 2   4 1 2   2 1 2 3     1 2 3   10 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

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

                3                   5     3                      11 2        
o16 = (map(R,R,{-x  + 4x  + x , x , -x  + -x  + x , x }), ideal (--x  + 4x x 
                8 1     2    4   1  9 1   2 2    3   2            8 1     1 2
      -----------------------------------------------------------------------
                   5 3     401 2 2       3   3 2           2     5 2      
      + x x  + 1, --x x  + ---x x  + 6x x  + -x x x  + 4x x x  + -x x x  +
         1 4      24 1 2   144 1 2     1 2   8 1 2 3     1 2 3   9 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

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,{x  + x , x , 4x  + 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, 4x x  + 2x x  + x x x  + 4x x x  + 2x x x  + x x x x  + 1), {x ,
           1 2     1 2    1 2 3     1 2 4     1 2 4    1 2 3 4         4 
      -----------------------------------------------------------------------
      x })
       3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :