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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

               10     7             1                            13 2   7    
o3 = (map(R,R,{--x  + -x  + x , x , -x  + 5x  + x , x }), ideal (--x  + -x x 
                3 1   3 2    4   1  2 1     2    3   2            3 1   3 1 2
     ------------------------------------------------------------------------
                 5 3     107 2 2   35   3   10 2       7   2     1 2      
     + x x  + 1, -x x  + ---x x  + --x x  + --x x x  + -x x x  + -x x x  +
        1 4      3 1 2    6  1 2    3 1 2    3 1 2 3   3 1 2 3   2 1 2 4  
     ------------------------------------------------------------------------
         2
     5x x x  + x x x x  + 1), {x , x })
       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

               7                        1         1     4                    
o6 = (map(R,R,{-x  + x  + x , x , 5x  + -x  + x , -x  + -x  + x , x }), ideal
               3 1    2    5   1    1   4 2    4  3 1   7 2    3   2         
     ------------------------------------------------------------------------
      7 2                  3  343 3     49 2 2   49 2           3        2  
     (-x  + x x  + x x  - x , ---x x  + --x x  + --x x x  + 7x x  + 14x x x 
      3 1    1 2    1 5    2   27 1 2    3 1 2    3 1 2 5     1 2      1 2 5
     ------------------------------------------------------------------------
             2    4     3       2 2      3
     + 7x x x  + x  + 3x x  + 3x x  + x x ), {x , x , x })
         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} | 21x_1x_2x_5^6-294x_2^9x_5-21x_2^9+147x_2^8x_5^2+21x_2
     {-9}  | 21x_1x_2^2x_5^3-147x_1x_2x_5^5+21x_1x_2x_5^4+2058x_2^
     {-9}  | 63x_1x_2^3+441x_1x_2^2x_5^2+126x_1x_2^2x_5+14406x_1x_
     {-3}  | 7x_1^2+3x_1x_2+3x_1x_5-3x_2^3                        
     ------------------------------------------------------------------------
                                                                            
     ^8x_5-49x_2^7x_5^3-21x_2^7x_5^2+21x_2^6x_5^3-21x_2^5x_5^4+21x_2^4x_5^5+
     9-1029x_2^8x_5-49x_2^8+343x_2^7x_5^2+98x_2^7x_5-147x_2^6x_5^2+147x_2^5x
     2x_5^5-1029x_1x_2x_5^4+294x_1x_2x_5^3+63x_1x_2x_5^2-201684x_2^9+100842x
                                                                            
     ------------------------------------------------------------------------
                                                                           
     9x_2^2x_5^6+9x_2x_5^7                                                 
     _5^3-147x_2^4x_5^4+21x_2^4x_5^3+9x_2^3x_5^3-63x_2^2x_5^5+18x_2^2x_5^4-
     _2^8x_5+7203x_2^8-33614x_2^7x_5^2-12005x_2^7x_5+343x_2^7+14406x_2^6x_5
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
     63x_2x_5^6+9x_2x_5^5                                                    
     ^2-1029x_2^6x_5-147x_2^6-14406x_2^5x_5^3+1029x_2^5x_5^2+147x_2^5x_5+63x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^5+14406x_2^4x_5^4-1029x_2^4x_5^3+294x_2^4x_5^2+63x_2^4x_5+27x_2^4+189x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _2^3x_5^2+81x_2^3x_5+6174x_2^2x_5^5-441x_2^2x_5^4+315x_2^2x_5^3+81x_2^2x
                                                                             
     ------------------------------------------------------------------------
                                                          |
                                                          |
                                                          |
     _5^2+6174x_2x_5^6-441x_2x_5^5+126x_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

                7      9                  5                      9 2    9    
o13 = (map(R,R,{-x  + --x  + x , x , x  + -x  + x , x }), ideal (-x  + --x x 
                2 1   10 2    4   1   1   3 2    3   2           2 1   10 1 2
      -----------------------------------------------------------------------
                  7 3     101 2 2   3   3   7 2        9   2      2      
      + x x  + 1, -x x  + ---x x  + -x x  + -x x x  + --x x x  + x x x  +
         1 4      2 1 2    15 1 2   2 1 2   2 1 2 3   10 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

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     8             2     9                      7 2   8    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                2 1   5 2    4   1  3 1   5 2    3   2           2 1   5 1 2
      -----------------------------------------------------------------------
                  5 3     167 2 2   72   3   5 2       8   2     2 2      
      + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      3 1 2    30 1 2   25 1 2   2 1 2 3   5 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      9   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

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