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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

               2     1             4     5                      5 2   1      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x  +
               3 1   2 2    4   1  9 1   6 2    3   2           3 1   2 1 2  
     ------------------------------------------------------------------------
                8 3     7 2 2    5   3   2 2       1   2     4 2      
     x x  + 1, --x x  + -x x  + --x x  + -x x x  + -x x x  + -x x x  +
      1 4      27 1 2   9 1 2   12 1 2   3 1 2 3   2 1 2 3   9 1 2 4  
     ------------------------------------------------------------------------
     5   2
     -x x x  + x x x x  + 1), {x , x })
     6 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     5             5     4         5     2              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , -x  + -x  + x , x }),
               5 1   2 2    5   1  8 1   3 2    4  4 1   5 2    3   2   
     ------------------------------------------------------------------------
            3 2   5               3   27 3     27 2 2   27 2       45   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
            5 1   2 1 2    1 5    2  125 1 2   10 1 2   25 1 2 5    4 1 2  
     ------------------------------------------------------------------------
         2     9     2   125 4   75 3     15 2 2      3
     9x x x  + -x x x  + ---x  + --x x  + --x x  + x x ), {x , x , x })
       1 2 5   5 1 2 5    8  2    4 2 5    2 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                         
     {-10} | 480x_1x_2x_5^6-10800x_2^9x_5-46875x_2^9+2160x_2^8x_5^2+18750x_2
     {-9}  | 93750x_1x_2^2x_5^3-4320x_1x_2x_5^5+37500x_1x_2x_5^4+97200x_2^9-
     {-9}  | 183105468750x_1x_2^3+8437500000x_1x_2^2x_5^2+146484375000x_1x_2
     {-3}  | 6x_1^2+25x_1x_2+10x_1x_5-10x_2^3                               
     ------------------------------------------------------------------------
                                                                          
     ^8x_5-288x_2^7x_5^3-7500x_2^7x_5^2+3000x_2^6x_5^3-1200x_2^5x_5^4+480x
     19440x_2^8x_5-56250x_2^8+2592x_2^7x_5^2+45000x_2^7x_5-27000x_2^6x_5^2
     ^2x_5+29859840x_1x_2x_5^5-129600000x_1x_2x_5^4+2250000000x_1x_2x_5^3+
                                                                          
     ------------------------------------------------------------------------
                                                                            
     _2^4x_5^5+2000x_2^2x_5^6+800x_2x_5^7                                   
     +10800x_2^5x_5^3-4320x_2^4x_5^4+37500x_2^4x_5^3+390625x_2^3x_5^3-18000x
     29296875000x_1x_2x_5^2-671846400x_2^9+134369280x_2^8x_5+583200000x_2^8-
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
     _2^2x_5^5+312500x_2^2x_5^4-7200x_2x_5^6+62500x_2x_5^5                   
     17915904x_2^7x_5^2-388800000x_2^7x_5+675000000x_2^7+186624000x_2^6x_5^2-
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     810000000x_2^6x_5-7031250000x_2^6-74649600x_2^5x_5^3+324000000x_2^5x_5^2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     +2812500000x_2^5x_5+73242187500x_2^5+29859840x_2^4x_5^4-129600000x_2^4x_
                                                                             
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     5^3+2250000000x_2^4x_5^2+29296875000x_2^4x_5+762939453125x_2^4+
                                                                    
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     35156250000x_2^3x_5^2+915527343750x_2^3x_5+124416000x_2^2x_5^5-
                                                                    
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     540000000x_2^2x_5^4+23437500000x_2^2x_5^3+366210937500x_2^2x_5^2+
                                                                      
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     49766400x_2x_5^6-216000000x_2x_5^5+3750000000x_2x_5^4+48828125000x_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                                               5 2          
o13 = (map(R,R,{-x  + 8x  + x , x , 3x  + x  + x , x }), ideal (-x  + 8x x  +
                2 1     2    4   1    1    2    3   2           2 1     1 2  
      -----------------------------------------------------------------------
                9 3     51 2 2       3   3 2           2       2          2
      x x  + 1, -x x  + --x x  + 8x x  + -x x x  + 8x x x  + 3x x x  + x x x 
       1 4      2 1 2    2 1 2     1 2   2 1 2 3     1 2 3     1 2 4    1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         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

                                    10     9                         2       
o16 = (map(R,R,{10x  + x  + x , x , --x  + -x  + x , x }), ideal (11x  + x x 
                   1    2    4   1   3 1   4 2    3   2              1    1 2
      -----------------------------------------------------------------------
                  100 3     155 2 2   9   3      2          2     10 2      
      + x x  + 1, ---x x  + ---x x  + -x x  + 10x x x  + x x x  + --x x x  +
         1 4       3  1 2    6  1 2   4 1 2      1 2 3    1 2 3    3 1 2 4  
      -----------------------------------------------------------------------
      9   2
      -x x x  + x x x x  + 1), {x , x })
      4 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

                                                                       2  
o19 = (map(R,R,{- 2x  - 2x  + x , x , - 2x  - x  + x , x }), ideal (- x  -
                    1     2    4   1      1    2    3   2              1  
      -----------------------------------------------------------------------
                          3       2 2       3     2           2       2      
      2x x  + x x  + 1, 4x x  + 6x x  + 2x x  - 2x x x  - 2x x x  - 2x x x  -
        1 2    1 4        1 2     1 2     1 2     1 2 3     1 2 3     1 2 4  
      -----------------------------------------------------------------------
         2
      x 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 :