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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

               7     3                                         12 2   3      
o3 = (map(R,R,{-x  + -x  + x , x , 5x  + x  + x , x }), ideal (--x  + -x x  +
               5 1   2 2    4   1    1    2    3   2            5 1   2 1 2  
     ------------------------------------------------------------------------
                 3     89 2 2   3   3   7 2       3   2       2          2
     x x  + 1, 7x x  + --x x  + -x x  + -x x x  + -x x x  + 5x x x  + x x x 
      1 4        1 2   10 1 2   2 1 2   5 1 2 3   2 1 2 3     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

                    1             2     7         1     4                    
o6 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , -x  + -x  + x , x }), ideal
                1   7 2    5   1  3 1   6 2    4  2 1   3 2    3   2         
     ------------------------------------------------------------------------
       2   1               3   3     3 2 2     2        3   3   6   2    
     (x  + -x x  + x x  - x , x x  + -x x  + 3x x x  + --x x  + -x x x  +
       1   7 1 2    1 5    2   1 2   7 1 2     1 2 5   49 1 2   7 1 2 5  
     ------------------------------------------------------------------------
           2    1  4    3 3     3 2 2      3
     3x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
       1 2 5   343 2   49 2 5   7 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                  
     {-10} | 16807x_1x_2x_5^6-2058x_2^9x_5-x_2^9+7203x_2^8x_5^2+7x_2^
     {-9}  | 7x_1x_2^2x_5^3-50421x_1x_2x_5^5+49x_1x_2x_5^4+6174x_2^9-
     {-9}  | 7x_1x_2^3+50421x_1x_2^2x_5^2+98x_1x_2^2x_5+249143169618x
     {-3}  | 7x_1^2+x_1x_2+7x_1x_5-7x_2^3                            
     ------------------------------------------------------------------------
                                                                    
     8x_5-16807x_2^7x_5^3-49x_2^7x_5^2+343x_2^6x_5^3-2401x_2^5x_5^4+
     21609x_2^8x_5-7x_2^8+50421x_2^7x_5^2+98x_2^7x_5-1029x_2^6x_5^2+
     _1x_2x_5^5-121060821x_1x_2x_5^4+235298x_1x_2x_5^3+343x_1x_2x_5^
                                                                    
     ------------------------------------------------------------------------
                                                                           
     16807x_2^4x_5^5+2401x_2^2x_5^6+16807x_2x_5^7                          
     7203x_2^5x_5^3-50421x_2^4x_5^4+49x_2^4x_5^3+x_2^3x_5^3-7203x_2^2x_5^5+
     2-30507326892x_2^9+106775644122x_2^8x_5+51883209x_2^8-249143169618x_2^
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
     14x_2^2x_5^4-50421x_2x_5^6+49x_2x_5^5                                   
     7x_5^2-605304105x_2^7x_5+117649x_2^7+5084554482x_2^6x_5^2-2470629x_2^6x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5-2401x_2^6-35591881374x_2^5x_5^3+17294403x_2^5x_5^2+16807x_2^5x_5+49x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^5+249143169618x_2^4x_5^4-121060821x_2^4x_5^3+235298x_2^4x_5^2+343x_2^4x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5+x_2^4+7203x_2^3x_5^2+21x_2^3x_5+35591881374x_2^2x_5^5-17294403x_2^2x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5^4+84035x_2^2x_5^3+147x_2^2x_5^2+249143169618x_2x_5^6-121060821x_2x_5^5
                                                                             
     ------------------------------------------------------------------------
                                 |
                                 |
                                 |
     +235298x_2x_5^4+343x_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

                5     9             8     10                      7 2   9    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (-x  + -x x 
                2 1   5 2    4   1  3 1    9 2    3   2           2 1   5 1 2
      -----------------------------------------------------------------------
                  20 3     341 2 2       3   5 2       9   2     8 2      
      + x x  + 1, --x x  + ---x x  + 2x x  + -x x x  + -x x x  + -x x x  +
         1 4       3 1 2    45 1 2     1 2   2 1 2 3   5 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      10   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

                7     3                                         16 2   3    
o16 = (map(R,R,{-x  + -x  + x , x , 3x  + x  + x , x }), ideal (--x  + -x x 
                9 1   4 2    4   1    1    2    3   2            9 1   4 1 2
      -----------------------------------------------------------------------
                  7 3     109 2 2   3   3   7 2       3   2       2      
      + x x  + 1, -x x  + ---x x  + -x x  + -x x x  + -x x x  + 3x x x  +
         1 4      3 1 2    36 1 2   4 1 2   9 1 2 3   4 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

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

                                                                       2  
o19 = (map(R,R,{- 4x  + 3x  + x , x , 2x  - 3x  + x , x }), ideal (- 3x  +
                    1     2    4   1    1     2    3   2               1  
      -----------------------------------------------------------------------
                            3        2 2       3     2           2    
      3x x  + x x  + 1, - 8x x  + 18x x  - 9x x  - 4x x x  + 3x x x  +
        1 2    1 4          1 2      1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      2x x x  - 3x x x  + x x x x  + 1), {x , x })
        1 2 4     1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :