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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

               1                   3     9                      5 2          
o3 = (map(R,R,{-x  + 8x  + x , x , -x  + -x  + x , x }), ideal (-x  + 8x x  +
               4 1     2    4   1  5 1   8 2    3   2           4 1     1 2  
     ------------------------------------------------------------------------
                3 3     813 2 2       3   1 2           2     3 2      
     x x  + 1, --x x  + ---x x  + 9x x  + -x x x  + 8x x x  + -x x x  +
      1 4      20 1 2   160 1 2     1 2   4 1 2 3     1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
     9   2
     -x x x  + x x x x  + 1), {x , x })
     8 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     9         1     2                    
o6 = (map(R,R,{x  + x  + x , x , -x  + -x  + x , -x  + -x  + x , x }), ideal
                1    2    5   1  5 1   8 2    4  2 1   5 2    3   2         
     ------------------------------------------------------------------------
       2                  3   3       2 2     2           3       2    
     (x  + x x  + x x  - x , x x  + 3x x  + 3x x x  + 3x x  + 6x x x  +
       1    1 2    1 5    2   1 2     1 2     1 2 5     1 2     1 2 5  
     ------------------------------------------------------------------------
           2    4     3       2 2      3
     3x 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} | x_1x_2x_5^6-6x_2^9x_5-x_2^9+3x_2^8x_5^2+x_2^8x_5-x_2^7x_5^3-x_2
     {-9}  | x_1x_2^2x_5^3-3x_1x_2x_5^5+x_1x_2x_5^4+18x_2^9-9x_2^8x_5-x_2^8+
     {-9}  | x_1x_2^3+3x_1x_2^2x_5^2+2x_1x_2^2x_5+18x_1x_2x_5^5-3x_1x_2x_5^4
     {-3}  | x_1^2+x_1x_2+x_1x_5-x_2^3                                      
     ------------------------------------------------------------------------
                                                                             
     ^7x_5^2+x_2^6x_5^3-x_2^5x_5^4+x_2^4x_5^5+x_2^2x_5^6+x_2x_5^7            
     3x_2^7x_5^2+2x_2^7x_5-3x_2^6x_5^2+3x_2^5x_5^3-3x_2^4x_5^4+x_2^4x_5^3+x_2
     +2x_1x_2x_5^3+x_1x_2x_5^2-108x_2^9+54x_2^8x_5+9x_2^8-18x_2^7x_5^2-15x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ^3x_5^3-3x_2^2x_5^5+2x_2^2x_5^4-3x_2x_5^6+x_2x_5^5                      
     7x_5+x_2^7+18x_2^6x_5^2-3x_2^6x_5-x_2^6-18x_2^5x_5^3+3x_2^5x_5^2+x_2^5x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5+x_2^5+18x_2^4x_5^4-3x_2^4x_5^3+2x_2^4x_5^2+x_2^4x_5+x_2^4+3x_2^3x_5^2+
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3x_2^3x_5+18x_2^2x_5^5-3x_2^2x_5^4+5x_2^2x_5^3+3x_2^2x_5^2+18x_2x_5^6-3x
                                                                             
     ------------------------------------------------------------------------
                                |
                                |
                                |
     _2x_5^5+2x_2x_5^4+x_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     7             1                           16 2   7    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , x }), ideal (--x  + -x x 
                9 1   3 2    4   1  3 1    2    3   2            9 1   3 1 2
      -----------------------------------------------------------------------
                   7 3     14 2 2   7   3   7 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      27 1 2    9 1 2   3 1 2   9 1 2 3   3 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
         2
      x x x  + x x x x  + 1), {x , x })
       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     3             1     7                      5 2   3    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                3 1   2 2    4   1  2 1   9 2    3   2           3 1   2 1 2
      -----------------------------------------------------------------------
                  1 3     137 2 2   7   3   2 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      3 1 2   108 1 2   6 1 2   3 1 2 3   2 1 2 3   2 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

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 :