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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

               4     5             1     4                      11 2   5    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
               7 1   8 2    4   1  5 1   5 2    3   2            7 1   8 1 2
     ------------------------------------------------------------------------
                  4 3     163 2 2   1   3   4 2       5   2     1 2      
     + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
        1 4      35 1 2   280 1 2   2 1 2   7 1 2 3   8 1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
     4   2
     -x x x  + x x x x  + 1), {x , x })
     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

               2                   1              9     7                    
o6 = (map(R,R,{-x  + 4x  + x , x , -x  + x  + x , -x  + -x  + x , x }), ideal
               3 1     2    5   1  3 1    2    4  4 1   6 2    3   2         
     ------------------------------------------------------------------------
      2 2                   3   8 3     16 2 2   4 2            3        2  
     (-x  + 4x x  + x x  - x , --x x  + --x x  + -x x x  + 32x x  + 16x 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
     + 2x x x  + 64x  + 48x x  + 12x 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} | 6x_1x_2x_5^6-384x_2^9x_5-6144x_2^9+48x_2^8x_5^2+1536x_2^8x_5-4x_
     {-9}  | 768x_1x_2^2x_5^3-6x_1x_2x_5^5+192x_1x_2x_5^4+384x_2^9-48x_2^8x_5
     {-9}  | 4718592x_1x_2^3+36864x_1x_2^2x_5^2+2359296x_1x_2^2x_5+6x_1x_2x_5
     {-3}  | 2x_1^2+12x_1x_2+3x_1x_5-3x_2^3                                  
     ------------------------------------------------------------------------
                                                                            
     2^7x_5^3-384x_2^7x_5^2+96x_2^6x_5^3-24x_2^5x_5^4+6x_2^4x_5^5+36x_2^2x_5
     -512x_2^8+4x_2^7x_5^2+256x_2^7x_5-96x_2^6x_5^2+24x_2^5x_5^3-6x_2^4x_5^4
     ^5-96x_1x_2x_5^4+6144x_1x_2x_5^3+294912x_1x_2x_5^2-384x_2^9+48x_2^8x_5+
                                                                            
     ------------------------------------------------------------------------
                                                                         
     ^6+9x_2x_5^7                                                        
     +192x_2^4x_5^3+4608x_2^3x_5^3-36x_2^2x_5^5+2304x_2^2x_5^4-9x_2x_5^6+
     768x_2^8-4x_2^7x_5^2-320x_2^7x_5+2048x_2^7+96x_2^6x_5^2-1536x_2^6x_5
                                                                         
     ------------------------------------------------------------------------
                                                                            
                                                                            
     288x_2x_5^5                                                            
     -49152x_2^6-24x_2^5x_5^3+384x_2^5x_5^2+12288x_2^5x_5+1179648x_2^5+6x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     4x_5^4-96x_2^4x_5^3+6144x_2^4x_5^2+294912x_2^4x_5+28311552x_2^4+221184x_
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     2^3x_5^2+21233664x_2^3x_5+36x_2^2x_5^5-576x_2^2x_5^4+92160x_2^2x_5^3+
                                                                          
     ------------------------------------------------------------------------
                                                                         |
                                                                         |
                                                                         |
     5308416x_2^2x_5^2+9x_2x_5^6-144x_2x_5^5+9216x_2x_5^4+442368x_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

                2     7                   10                      5 2   7    
o13 = (map(R,R,{-x  + -x  + x , x , 3x  + --x  + x , x }), ideal (-x  + -x x 
                3 1   6 2    4   1    1    3 2    3   2           3 1   6 1 2
      -----------------------------------------------------------------------
                    3     103 2 2   35   3   2 2       7   2       2      
      + x x  + 1, 2x x  + ---x x  + --x x  + -x x x  + -x x x  + 3x x x  +
         1 4        1 2    18 1 2    9 1 2   3 1 2 3   6 1 2 3     1 2 4  
      -----------------------------------------------------------------------
      10   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

                4     3             1     6                      11 2   3    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                7 1   2 2    4   1  2 1   7 2    3   2            7 1   2 1 2
      -----------------------------------------------------------------------
                  2 3     243 2 2   9   3   4 2       3   2     1 2      
      + x x  + 1, -x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      7 1 2   196 1 2   7 1 2   7 1 2 3   2 1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      6   2
      -x x x  + x x x x  + 1), {x , x })
      7 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
--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: 3
--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: 4
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                 2          
o19 = (map(R,R,{x  + 4x  + x , x , x  - 8x  + x , x }), ideal (2x  + 4x x  +
                 1     2    4   1   1     2    3   2             1     1 2  
      -----------------------------------------------------------------------
                 3       2 2        3    2           2      2           2
      x x  + 1, x x  - 4x x  - 32x x  + x x x  + 4x x x  + x x x  - 8x 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 :