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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

               7     8             3     7                      12 2   8    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
               5 1   7 2    4   1  5 1   9 2    3   2            5 1   7 1 2
     ------------------------------------------------------------------------
                 21 3     559 2 2   8   3   7 2       8   2     3 2      
     + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
        1 4      25 1 2   315 1 2   9 1 2   5 1 2 3   7 1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
     7   2
     -x x x  + x x x x  + 1), {x , x })
     9 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                                 3     3                    
o6 = (map(R,R,{-x  + x  + x , x , 8x  + x  + x , -x  + -x  + x , x }), ideal
               9 1    2    5   1    1    2    4  5 1   5 2    3   2         
     ------------------------------------------------------------------------
      8 2                  3  512 3     64 2 2   64 2       8   3   16   2  
     (-x  + x x  + x x  - x , ---x x  + --x x  + --x x x  + -x x  + --x x x 
      9 1    1 2    1 5    2  729 1 2   27 1 2   27 1 2 5   3 1 2    3 1 2 5
     ------------------------------------------------------------------------
       8     2    4     3       2 2      3
     + -x x x  + x  + 3x x  + 3x x  + x x ), {x , x , x })
       3 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} | 72x_1x_2x_5^6-384x_2^9x_5-72x_2^9+192x_2^8x_5^2+72x_2^8x_5-64x_2
     {-9}  | 216x_1x_2^2x_5^3-576x_1x_2x_5^5+216x_1x_2x_5^4+3072x_2^9-1536x_2
     {-9}  | 5832x_1x_2^3+15552x_1x_2^2x_5^2+11664x_1x_2^2x_5+73728x_1x_2x_5^
     {-3}  | 8x_1^2+9x_1x_2+9x_1x_5-9x_2^3                                   
     ------------------------------------------------------------------------
                                                                     
     ^7x_5^3-72x_2^7x_5^2+72x_2^6x_5^3-72x_2^5x_5^4+72x_2^4x_5^5+81x_
     ^8x_5-192x_2^8+512x_2^7x_5^2+384x_2^7x_5-576x_2^6x_5^2+576x_2^5x
     5-13824x_1x_2x_5^4+10368x_1x_2x_5^3+5832x_1x_2x_5^2-393216x_2^9+
                                                                     
     ------------------------------------------------------------------------
                                                                             
     2^2x_5^6+81x_2x_5^7                                                     
     _5^3-576x_2^4x_5^4+216x_2^4x_5^3+243x_2^3x_5^3-648x_2^2x_5^5+486x_2^2x_5
     196608x_2^8x_5+36864x_2^8-65536x_2^7x_5^2-61440x_2^7x_5+4608x_2^7+73728x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ^4-648x_2x_5^6+243x_2x_5^5                                              
     _2^6x_5^2-13824x_2^6x_5-5184x_2^6-73728x_2^5x_5^3+13824x_2^5x_5^2+5184x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^5x_5+5832x_2^5+73728x_2^4x_5^4-13824x_2^4x_5^3+10368x_2^4x_5^2+5832x_2
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     ^4x_5+6561x_2^4+17496x_2^3x_5^2+19683x_2^3x_5+82944x_2^2x_5^5-15552x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     2x_5^4+29160x_2^2x_5^3+19683x_2^2x_5^2+82944x_2x_5^6-15552x_2x_5^5+
                                                                        
     ------------------------------------------------------------------------
                                |
                                |
                                |
     11664x_2x_5^4+6561x_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                   1                        2   3      
o13 = (map(R,R,{x  + -x  + x , x , 3x  + -x  + x , x }), ideal (2x  + -x x  +
                 1   5 2    4   1    1   2 2    3   2             1   5 1 2  
      -----------------------------------------------------------------------
                  3     23 2 2    3   3    2       3   2       2      
      x x  + 1, 3x x  + --x x  + --x x  + x x x  + -x x x  + 3x x x  +
       1 4        1 2   10 1 2   10 1 2    1 2 3   5 1 2 3     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

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                   7     9                      7 2        
o16 = (map(R,R,{-x  + 2x  + x , x , -x  + -x  + x , x }), ideal (-x  + 2x x 
                2 1     2    4   1  4 1   8 2    3   2           2 1     1 2
      -----------------------------------------------------------------------
                  35 3     101 2 2   9   3   5 2           2     7 2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + 2x x x  + -x x x  +
         1 4       8 1 2    16 1 2   4 1 2   2 1 2 3     1 2 3   4 1 2 4  
      -----------------------------------------------------------------------
      9   2
      -x x x  + x x x x  + 1), {x , x })
      8 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,{- 3x  - 3x  + x , x , - x  - 2x  + x , x }), ideal (- 2x  -
                    1     2    4   1     1     2    3   2               1  
      -----------------------------------------------------------------------
                          3       2 2       3     2           2      2      
      3x x  + x x  + 1, 3x x  + 9x x  + 6x x  - 3x x x  - 3x x x  - x x x  -
        1 2    1 4        1 2     1 2     1 2     1 2 3     1 2 3    1 2 4  
      -----------------------------------------------------------------------
          2
      2x x x  + x x x x  + 1), {x , x })
        1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :