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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

                     3              9     8                        2   3    
o3 = (map(R,R,{2x  + -x  + x , x , --x  + -x  + x , x }), ideal (3x  + -x x 
                 1   5 2    4   1  10 1   5 2    3   2             1   5 1 2
     ------------------------------------------------------------------------
                 9 3     187 2 2   24   3     2       3   2      9 2      
     + x x  + 1, -x x  + ---x x  + --x x  + 2x x x  + -x x x  + --x x x  +
        1 4      5 1 2    50 1 2   25 1 2     1 2 3   5 1 2 3   10 1 2 4  
     ------------------------------------------------------------------------
     8   2
     -x x x  + x x x x  + 1), {x , x })
     5 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     1             8              4     2                    
o6 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , -x  + -x  + x , x }), ideal
               3 1   3 2    5   1  7 1    2    4  9 1   3 2    3   2         
     ------------------------------------------------------------------------
      1 2   1               3   1 3     1 2 2   1 2       1   3   2   2    
     (-x  + -x x  + x x  - x , --x x  + -x x  + -x x x  + -x x  + -x x x  +
      3 1   3 1 2    1 5    2  27 1 2   9 1 2   3 1 2 5   9 1 2   3 1 2 5  
     ------------------------------------------------------------------------
          2    1 4   1 3      2 2      3
     x x x  + --x  + -x x  + x x  + x x ), {x , x , x })
      1 2 5   27 2   3 2 5    2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 243x_1x_2x_5^6-54x_2^9x_5-x_2^9+81x_2^8x_5^2+3x_2^8x_5-81x_2^7x_
     {-9}  | 3x_1x_2^2x_5^3-243x_1x_2x_5^5+9x_1x_2x_5^4+54x_2^9-81x_2^8x_5-x_
     {-9}  | x_1x_2^3+81x_1x_2^2x_5^2+6x_1x_2^2x_5+118098x_1x_2x_5^5-2187x_1x
     {-3}  | x_1^2+x_1x_2+3x_1x_5-3x_2^3                                     
     ------------------------------------------------------------------------
                                                                           
     5^3-9x_2^7x_5^2+27x_2^6x_5^3-81x_2^5x_5^4+243x_2^4x_5^5+243x_2^2x_5^6+
     2^8+81x_2^7x_5^2+6x_2^7x_5-27x_2^6x_5^2+81x_2^5x_5^3-243x_2^4x_5^4+9x_
     _2x_5^4+162x_1x_2x_5^3+9x_1x_2x_5^2-26244x_2^9+39366x_2^8x_5+729x_2^8-
                                                                           
     ------------------------------------------------------------------------
                                                                             
     729x_2x_5^7                                                             
     2^4x_5^3+3x_2^3x_5^3-243x_2^2x_5^5+18x_2^2x_5^4-729x_2x_5^6+27x_2x_5^5  
     39366x_2^7x_5^2-3645x_2^7x_5+27x_2^7+13122x_2^6x_5^2-243x_2^6x_5-9x_2^6-
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     39366x_2^5x_5^3+729x_2^5x_5^2+27x_2^5x_5+3x_2^5+118098x_2^4x_5^4-2187x_2
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     ^4x_5^3+162x_2^4x_5^2+9x_2^4x_5+x_2^4+81x_2^3x_5^2+9x_2^3x_5+118098x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2x_5^5-2187x_2^2x_5^4+405x_2^2x_5^3+27x_2^2x_5^2+354294x_2x_5^6-6561x_2x
                                                                             
     ------------------------------------------------------------------------
                                 |
                                 |
                                 |
     _5^5+486x_2x_5^4+27x_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

                     9                                             2   9    
o13 = (map(R,R,{x  + -x  + x , x , 3x  + 10x  + x , x }), ideal (2x  + -x x 
                 1   2 2    4   1    1      2    3   2             1   2 1 2
      -----------------------------------------------------------------------
                    3     47 2 2        3    2       9   2       2      
      + x x  + 1, 3x x  + --x x  + 45x x  + x x x  + -x x x  + 3x x x  +
         1 4        1 2    2 1 2      1 2    1 2 3   2 1 2 3     1 2 4  
      -----------------------------------------------------------------------
           2
      10x 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

                7     4             9     8                      11 2   4    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                4 1   5 2    4   1  5 1   9 2    3   2            4 1   5 1 2
      -----------------------------------------------------------------------
                  63 3     674 2 2   32   3   7 2       4   2     9 2      
      + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      20 1 2   225 1 2   45 1 2   4 1 2 3   5 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      8   2
      -x x x  + x x x x  + 1), {x , x })
      9 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,{2x  - x  + x , x , - 4x  - 5x  + x , x }), ideal (3x  - x x 
                  1    2    4   1      1     2    3   2             1    1 2
      -----------------------------------------------------------------------
                      3       2 2       3     2          2       2      
      + x x  + 1, - 8x x  - 6x x  + 5x x  + 2x x x  - x x x  - 4x x x  -
         1 4          1 2     1 2     1 2     1 2 3    1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      5x 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 :