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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

               1     10             2      1                      8 2  
o3 = (map(R,R,{-x  + --x  + x , x , -x  + --x  + x , x }), ideal (-x  +
               7 1    7 2    4   1  3 1   10 2    3   2           7 1  
     ------------------------------------------------------------------------
     10                  2 3     29 2 2   1   3   1 2       10   2    
     --x x  + x x  + 1, --x x  + --x x  + -x x  + -x x x  + --x x x  +
      7 1 2    1 4      21 1 2   30 1 2   7 1 2   7 1 2 3    7 1 2 3  
     ------------------------------------------------------------------------
     2 2        1   2
     -x x x  + --x x x  + x x x x  + 1), {x , x })
     3 1 2 4   10 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      9             3              1     1              
o6 = (map(R,R,{-x  + --x  + x , x , -x  + x  + x , -x  + -x  + x , x }),
               2 1   10 2    5   1  8 1    2    4  5 1   2 2    3   2   
     ------------------------------------------------------------------------
            3 2    9               3  27 3     243 2 2   27 2       729   3  
     ideal (-x  + --x x  + x x  - x , --x x  + ---x x  + --x x x  + ---x x  +
            2 1   10 1 2    1 5    2   8 1 2    40 1 2    4 1 2 5   200 1 2  
     ------------------------------------------------------------------------
     81   2     9     2    729 4   243 3     27 2 2      3
     --x x x  + -x x x  + ----x  + ---x x  + --x x  + x x ), {x , x , x })
     10 1 2 5   2 1 2 5   1000 2   100 2 5   10 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 300000x_1x_2x_5^6-2187000x_2^9x_5-177147x_2^9+1215000x_2^8x_5^2+
     {-9}  | 21870x_1x_2^2x_5^3-150000x_1x_2x_5^5+24300x_1x_2x_5^4+1093500x_2
     {-9}  | 1937102445x_1x_2^3+13286025000x_1x_2^2x_5^2+4304672100x_1x_2^2x_
     {-3}  | 15x_1^2+9x_1x_2+10x_1x_5-10x_2^3                                
     ------------------------------------------------------------------------
                                                   
     196830x_2^8x_5-450000x_2^7x_5^3-218700x_2^7x_5
     ^9-607500x_2^8x_5-32805x_2^8+225000x_2^7x_5^2+
     5+375000000000x_1x_2x_5^5-30375000000x_1x_2x_5
                                                   
     ------------------------------------------------------------------------
                                                                     
     ^2+243000x_2^6x_5^3-270000x_2^5x_5^4+300000x_2^4x_5^5+180000x_2^
     72900x_2^7x_5-121500x_2^6x_5^2+135000x_2^5x_5^3-150000x_2^4x_5^4
     ^4+9841500000x_1x_2x_5^3+2391484500x_1x_2x_5^2-2733750000000x_2^
                                                                     
     ------------------------------------------------------------------------
                                                                      
     2x_5^6+200000x_2x_5^7                                            
     +24300x_2^4x_5^3+13122x_2^3x_5^3-90000x_2^2x_5^5+29160x_2^2x_5^4-
     9+1518750000000x_2^8x_5+123018750000x_2^8-562500000000x_2^7x_5^2-
                                                                      
     ------------------------------------------------------------------------
                                                                             
                                                                             
     100000x_2x_5^6+16200x_2x_5^5                                            
     227812500000x_2^7x_5+7381125000x_2^7+303750000000x_2^6x_5^2-24603750000x
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     _2^6x_5-3985807500x_2^6-337500000000x_2^5x_5^3+27337500000x_2^5x_5^2+
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     4428675000x_2^5x_5+2152336050x_2^5+375000000000x_2^4x_5^4-30375000000x_2
                                                                             
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     ^4x_5^3+9841500000x_2^4x_5^2+2391484500x_2^4x_5+1162261467x_2^4+
                                                                     
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     7971615000x_2^3x_5^2+3874204890x_2^3x_5+225000000000x_2^2x_5^5-
                                                                    
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     18225000000x_2^2x_5^4+14762250000x_2^2x_5^3+4304672100x_2^2x_5^2+
                                                                      
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     250000000000x_2x_5^6-20250000000x_2x_5^5+6561000000x_2x_5^4+1594323000x_
                                                                             
     ------------------------------------------------------------------------
            |
            |
            |
     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

                     1             6     7                        2   1      
o13 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , x }), ideal (2x  + -x x  +
                 1   9 2    4   1  5 1   9 2    3   2             1   9 1 2  
      -----------------------------------------------------------------------
                6 3     41 2 2    7   3    2       1   2     6 2      
      x x  + 1, -x x  + --x x  + --x x  + x x x  + -x x x  + -x x x  +
       1 4      5 1 2   45 1 2   81 1 2    1 2 3   9 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

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

                     2             4     4                        2   2      
o16 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , x }), ideal (2x  + -x x  +
                 1   3 2    4   1  9 1   9 2    3   2             1   3 1 2  
      -----------------------------------------------------------------------
                4 3     20 2 2    8   3    2       2   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   27 1 2    1 2 3   3 1 2 3   9 1 2 4  
      -----------------------------------------------------------------------
      4   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

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

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :