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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

                     83                   28                        2  
o3 = (map(R,R,{2x  + --x  + x , x , 2x  + --x  + x , x }), ideal (3x  +
                 1    8 2    4   1    1    5 2    3   2             1  
     ------------------------------------------------------------------------
     83                   3     639 2 2   581   3     2       83   2    
     --x x  + x x  + 1, 4x x  + ---x x  + ---x x  + 2x x x  + --x x x  +
      8 1 2    1 4        1 2    20 1 2    10 1 2     1 2 3    8 1 2 3  
     ------------------------------------------------------------------------
       2       28   2
     2x x x  + --x x x  + x x x x  + 1), {x , x })
       1 2 4    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                    9               4      3              
o6 = (map(R,R,{-x  + 10x  + x , x , -x  + 4x  + x , -x  + --x  + x , x }),
               2 1      2    5   1  7 1     2    4  9 1   10 2    3   2   
     ------------------------------------------------------------------------
            1 2                    3  1 3     15 2 2   3 2             3  
     ideal (-x  + 10x x  + x x  - x , -x x  + --x x  + -x x x  + 150x x  +
            2 1      1 2    1 5    2  8 1 2    2 1 2   4 1 2 5       1 2  
     ------------------------------------------------------------------------
          2     3     2        4       3        2 2      3
     30x x x  + -x x x  + 1000x  + 300x x  + 30x x  + x x ), {x , x , x })
        1 2 5   2 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} | 2x_1x_2x_5^6-600x_2^9x_5-200000x_2^9+30x_2^8x_5^2+20000x_2^8x_5
     {-9}  | 40000x_1x_2^2x_5^3-6x_1x_2x_5^5+4000x_1x_2x_5^4+1800x_2^9-90x_2
     {-9}  | 800000000000x_1x_2^3+120000000x_1x_2^2x_5^2+160000000000x_1x_2^
     {-3}  | x_1^2+20x_1x_2+2x_1x_5-2x_2^3                                  
     ------------------------------------------------------------------------
                                                                             
     -x_2^7x_5^3-2000x_2^7x_5^2+200x_2^6x_5^3-20x_2^5x_5^4+2x_2^4x_5^5+40x_2^
     ^8x_5-20000x_2^8+3x_2^7x_5^2+4000x_2^7x_5-600x_2^6x_5^2+60x_2^5x_5^3-6x_
     2x_5+18x_1x_2x_5^5-6000x_1x_2x_5^4+8000000x_1x_2x_5^3+8000000000x_1x_2x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
     2x_5^6+4x_2x_5^7                                                        
     2^4x_5^4+4000x_2^4x_5^3+800000x_2^3x_5^3-120x_2^2x_5^5+160000x_2^2x_5^4-
     5^2-5400x_2^9+270x_2^8x_5+90000x_2^8-9x_2^7x_5^2-15000x_2^7x_5+2000000x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     12x_2x_5^6+8000x_2x_5^5                                                 
     2^7+1800x_2^6x_5^2-600000x_2^6x_5-400000000x_2^6-180x_2^5x_5^3+60000x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     5x_5^2+40000000x_2^5x_5+80000000000x_2^5+18x_2^4x_5^4-6000x_2^4x_5^3+
                                                                          
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     8000000x_2^4x_5^2+8000000000x_2^4x_5+16000000000000x_2^4+2400000000x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3x_5^2+4800000000000x_2^3x_5+360x_2^2x_5^5-120000x_2^2x_5^4+400000000x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2x_5^3+480000000000x_2^2x_5^2+36x_2x_5^6-12000x_2x_5^5+16000000x_2x_5^4
                                                                             
     ------------------------------------------------------------------------
                          |
                          |
                          |
     +16000000000x_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

                       3             1     1                         2  
o13 = (map(R,R,{10x  + -x  + x , x , -x  + -x  + x , x }), ideal (11x  +
                   1   5 2    4   1  6 1   7 2    3   2              1  
      -----------------------------------------------------------------------
      3                 5 3     107 2 2    3   3      2       3   2    
      -x x  + x x  + 1, -x x  + ---x x  + --x x  + 10x x x  + -x x x  +
      5 1 2    1 4      3 1 2    70 1 2   35 1 2      1 2 3   5 1 2 3  
      -----------------------------------------------------------------------
      1 2       1   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      6 1 2 4   7 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     4             4     6                      8 2   4    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                3 1   9 2    4   1  3 1   5 2    3   2           3 1   9 1 2
      -----------------------------------------------------------------------
                  20 3     70 2 2    8   3   5 2       4   2     4 2      
      + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4       9 1 2   27 1 2   15 1 2   3 1 2 3   9 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      6   2
      -x x x  + x x x x  + 1), {x , x })
      5 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

                                                         2                 
o19 = (map(R,R,{- 5x  + x , x , - x  + x , x }), ideal (x  - 5x x  + x x  +
                    2    4   1     2    3   2            1     1 2    1 4  
      -----------------------------------------------------------------------
             3       2        2
      1, 5x x  - 5x x x  - x x x  + x x x x  + 1), {x , x })
           1 2     1 2 3    1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :