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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     3             3     1                      5 2   3      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x  +
               2 1   4 2    4   1  4 1   9 2    3   2           2 1   4 1 2  
     ------------------------------------------------------------------------
               9 3     35 2 2    1   3   3 2       3   2     3 2      
     x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
      1 4      8 1 2   48 1 2   12 1 2   2 1 2 3   4 1 2 3   4 1 2 4  
     ------------------------------------------------------------------------
     1   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

                     3             1     4          3     4              
o6 = (map(R,R,{4x  + -x  + x , x , -x  + -x  + x , --x  + -x  + x , x }),
                 1   4 2    5   1  3 1   3 2    4  10 1   5 2    3   2   
     ------------------------------------------------------------------------
              2   3               3     3        2 2      2       27   3  
     ideal (4x  + -x x  + x x  - x , 64x x  + 36x x  + 48x x x  + --x x  +
              1   4 1 2    1 5    2     1 2      1 2      1 2 5    4 1 2  
     ------------------------------------------------------------------------
          2            2   27 4   27 3     9 2 2      3
     18x x x  + 12x x x  + --x  + --x x  + -x x  + x x ), {x , x , x })
        1 2 5      1 2 5   64 2   16 2 5   4 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                           
     {-10} | 1024x_1x_2x_5^6-13824x_2^9x_5-243x_2^9+9216x_2^8x
     {-9}  | 432x_1x_2^2x_5^3-16384x_1x_2x_5^5+576x_1x_2x_5^4+
     {-9}  | 314928x_1x_2^3+11943936x_1x_2^2x_5^2+839808x_1x_2
     {-3}  | 16x_1^2+3x_1x_2+4x_1x_5-4x_2^3                   
     ------------------------------------------------------------------------
                                                                       
     _5^2+324x_2^8x_5-4096x_2^7x_5^3-432x_2^7x_5^2+576x_2^6x_5^3-768x_2
     221184x_2^9-147456x_2^8x_5-1728x_2^8+65536x_2^7x_5^2+4608x_2^7x_5-
     ^2x_5+8589934592x_1x_2x_5^5-150994944x_1x_2x_5^4+10616832x_1x_2x_5
                                                                       
     ------------------------------------------------------------------------
                                                                             
     ^5x_5^4+1024x_2^4x_5^5+192x_2^2x_5^6+256x_2x_5^7                        
     9216x_2^6x_5^2+12288x_2^5x_5^3-16384x_2^4x_5^4+576x_2^4x_5^3+81x_2^3x_5^
     ^3+559872x_1x_2x_5^2-115964116992x_2^9+77309411328x_2^8x_5+1358954496x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     3-3072x_2^2x_5^5+216x_2^2x_5^4-4096x_2x_5^6+144x_2x_5^5                 
     ^8-34359738368x_2^7x_5^2-3019898880x_2^7x_5+21233664x_2^7+4831838208x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     6x_5^2-84934656x_2^6x_5-2985984x_2^6-6442450944x_2^5x_5^3+113246208x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5x_5^2+3981312x_2^5x_5+419904x_2^5+8589934592x_2^4x_5^4-150994944x_2^4x_
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     5^3+10616832x_2^4x_5^2+559872x_2^4x_5+59049x_2^4+2239488x_2^3x_5^2+
                                                                        
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     236196x_2^3x_5+1610612736x_2^2x_5^5-28311552x_2^2x_5^4+4976640x_2^2x_5^3
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     +314928x_2^2x_5^2+2147483648x_2x_5^6-37748736x_2x_5^5+2654208x_2x_5^4+
                                                                           
     ------------------------------------------------------------------------
                    |
                    |
                    |
     139968x_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     9             1      7                      4 2   9    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (-x  + -x x 
                3 1   4 2    4   1  5 1   10 2    3   2           3 1   4 1 2
      -----------------------------------------------------------------------
                   1 3     41 2 2   63   3   1 2       9   2     1 2      
      + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      15 1 2   60 1 2   40 1 2   3 1 2 3   4 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
       7   2
      --x x x  + x x x x  + 1), {x , x })
      10 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     1                   5                      11 2   1    
o16 = (map(R,R,{-x  + -x  + x , x , 3x  + -x  + x , x }), ideal (--x  + -x x 
                4 1   4 2    4   1    1   3 2    3   2            4 1   4 1 2
      -----------------------------------------------------------------------
                  21 3     11 2 2    5   3   7 2       1   2       2      
      + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + 3x x x  +
         1 4       4 1 2    3 1 2   12 1 2   4 1 2 3   4 1 2 3     1 2 4  
      -----------------------------------------------------------------------
      5   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 , - 5x  - 4x  + x , x }), ideal (- 3x  +
                    1     2    4   1      1     2    3   2               1  
      -----------------------------------------------------------------------
                           3       2 2        3     2           2       2    
      4x x  + x x  + 1, 20x x  - 4x x  - 16x x  - 4x x x  + 4x x x  - 5x x x 
        1 2    1 4         1 2     1 2      1 2     1 2 3     1 2 3     1 2 4
      -----------------------------------------------------------------------
            2
      - 4x 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 :