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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     1             3                           4 2   1      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , x }), ideal (-x  + -x x  +
               3 1   8 2    4   1  8 1    2    3   2           3 1   8 1 2  
     ------------------------------------------------------------------------
               1 3      73 2 2   1   3   1 2       1   2     3 2          2
     x x  + 1, -x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  + x x x 
      1 4      8 1 2   192 1 2   8 1 2   3 1 2 3   8 1 2 3   8 1 2 4    1 2 4
     ------------------------------------------------------------------------
     + x x x x  + 1), {x , x })
        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

                                  5      3         1                    
o6 = (map(R,R,{x  + 3x  + x , x , -x  + --x  + x , -x  + 2x  + x , x }),
                1     2    5   1  9 1   10 2    4  4 1     2    3   2   
     ------------------------------------------------------------------------
             2                   3   3       2 2     2            3  
     ideal (x  + 3x x  + x x  - x , x x  + 9x x  + 3x x x  + 27x x  +
             1     1 2    1 5    2   1 2     1 2     1 2 5      1 2  
     ------------------------------------------------------------------------
          2           2      4      3       2 2      3
     18x x x  + 3x x x  + 27x  + 27x x  + 9x x  + x x ), {x , x , x })
        1 2 5     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} | x_1x_2x_5^6-54x_2^9x_5-243x_2^9+9x_2^8x_5^2+81x_2^8x_5-x_2^7x_5^
     {-9}  | 27x_1x_2^2x_5^3-x_1x_2x_5^5+9x_1x_2x_5^4+54x_2^9-9x_2^8x_5-27x_2
     {-9}  | 19683x_1x_2^3+729x_1x_2^2x_5^2+13122x_1x_2^2x_5+2x_1x_2x_5^5-9x_
     {-3}  | x_1^2+3x_1x_2+x_1x_5-x_2^3                                      
     ------------------------------------------------------------------------
                                                                             
     3-27x_2^7x_5^2+9x_2^6x_5^3-3x_2^5x_5^4+x_2^4x_5^5+3x_2^2x_5^6+x_2x_5^7  
     ^8+x_2^7x_5^2+18x_2^7x_5-9x_2^6x_5^2+3x_2^5x_5^3-x_2^4x_5^4+9x_2^4x_5^3+
     1x_2x_5^4+162x_1x_2x_5^3+2187x_1x_2x_5^2-108x_2^9+18x_2^8x_5+81x_2^8-2x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     81x_2^3x_5^3-3x_2^2x_5^5+54x_2^2x_5^4-x_2x_5^6+9x_2x_5^5                
     2^7x_5^2-45x_2^7x_5+81x_2^7+18x_2^6x_5^2-81x_2^6x_5-729x_2^6-6x_2^5x_5^3
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     +27x_2^5x_5^2+243x_2^5x_5+6561x_2^5+2x_2^4x_5^4-9x_2^4x_5^3+162x_2^4x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2+2187x_2^4x_5+59049x_2^4+2187x_2^3x_5^2+59049x_2^3x_5+6x_2^2x_5^5-27x_2
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     ^2x_5^4+1215x_2^2x_5^3+19683x_2^2x_5^2+2x_2x_5^6-9x_2x_5^5+162x_2x_5^4+
                                                                            
     ------------------------------------------------------------------------
                  |
                  |
                  |
     2187x_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

                7     1             9     7                      10 2   1    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                3 1   3 2    4   1  2 1   3 2    3   2            3 1   3 1 2
      -----------------------------------------------------------------------
                  21 3     125 2 2   7   3   7 2       1   2     9 2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4       2 1 2    18 1 2   9 1 2   3 1 2 3   3 1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      7   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

                2                  5                            9 2         
o16 = (map(R,R,{-x  + x  + x , x , -x  + 2x  + x , x }), ideal (-x  + x x  +
                7 1    2    4   1  7 1     2    3   2           7 1    1 2  
      -----------------------------------------------------------------------
                10 3     9 2 2       3   2 2          2     5 2           2
      x x  + 1, --x x  + -x x  + 2x x  + -x x x  + x x x  + -x x x  + 2x x x 
       1 4      49 1 2   7 1 2     1 2   7 1 2 3    1 2 3   7 1 2 4     1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         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

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