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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                                            2    1    
o3 = (map(R,R,{x  + --x  + x , x , 8x  + 3x  + x , x }), ideal (2x  + --x x 
                1   10 2    4   1    1     2    3   2             1   10 1 2
     ------------------------------------------------------------------------
                   3     19 2 2    3   3    2        1   2       2      
     + x x  + 1, 8x x  + --x x  + --x x  + x x x  + --x x x  + 8x x x  +
        1 4        1 2    5 1 2   10 1 2    1 2 3   10 1 2 3     1 2 4  
     ------------------------------------------------------------------------
         2
     3x 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

               9                                 7                          
o6 = (map(R,R,{-x  + 9x  + x , x , x  + x  + x , -x  + 9x  + x , x }), ideal
               5 1     2    5   1   1    2    4  8 1     2    3   2         
     ------------------------------------------------------------------------
      9 2                   3  729 3     2187 2 2   243 2       2187   3  
     (-x  + 9x x  + x x  - x , ---x x  + ----x x  + ---x x x  + ----x x  +
      5 1     1 2    1 5    2  125 1 2    25  1 2    25 1 2 5     5  1 2  
     ------------------------------------------------------------------------
     486   2     27     2       4       3        2 2      3
     ---x x x  + --x x x  + 729x  + 243x x  + 27x x  + x x ), {x , x , x })
      5  1 2 5    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} | 45x_1x_2x_5^6-39366x_2^9x_5-2657205x_2^9+2187x_2^8x_5^2+
     {-9}  | 54675x_1x_2^2x_5^3-45x_1x_2x_5^5+6075x_1x_2x_5^4+39366x_
     {-9}  | 26904200625x_1x_2^3+22143375x_1x_2^2x_5^2+5978711250x_1x
     {-3}  | 9x_1^2+45x_1x_2+5x_1x_5-5x_2^3                          
     ------------------------------------------------------------------------
                                                                           
     295245x_2^8x_5-81x_2^7x_5^3-32805x_2^7x_5^2+3645x_2^6x_5^3-405x_2^5x_5
     2^9-2187x_2^8x_5-98415x_2^8+81x_2^7x_5^2+21870x_2^7x_5-3645x_2^6x_5^2+
     _2^2x_5+90x_1x_2x_5^5-6075x_1x_2x_5^4+1640250x_1x_2x_5^3+332150625x_1x
                                                                           
     ------------------------------------------------------------------------
                                                                             
     ^4+45x_2^4x_5^5+225x_2^2x_5^6+25x_2x_5^7                                
     405x_2^5x_5^3-45x_2^4x_5^4+6075x_2^4x_5^3+273375x_2^3x_5^3-225x_2^2x_5^5
     _2x_5^2-78732x_2^9+4374x_2^8x_5+295245x_2^8-162x_2^7x_5^2-54675x_2^7x_5+
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
     +60750x_2^2x_5^4-25x_2x_5^6+3375x_2x_5^5                               
     1476225x_2^7+7290x_2^6x_5^2-492075x_2^6x_5-66430125x_2^6-810x_2^5x_5^3+
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     54675x_2^5x_5^2+7381125x_2^5x_5+2989355625x_2^5+90x_2^4x_5^4-6075x_2^4x_
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     5^3+1640250x_2^4x_5^2+332150625x_2^4x_5+134521003125x_2^4+110716875x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3x_5^2+44840334375x_2^3x_5+450x_2^2x_5^5-30375x_2^2x_5^4+20503125x_2^2x_
                                                                             
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     5^3+4982259375x_2^2x_5^2+50x_2x_5^6-3375x_2x_5^5+911250x_2x_5^4+
                                                                     
     ------------------------------------------------------------------------
                       |
                       |
                       |
     184528125x_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

                 1     3             5     6                      11 2  
o13 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  +
                10 1   2 2    4   1  3 1   7 2    3   2           10 1  
      -----------------------------------------------------------------------
      3                 1 3     181 2 2   9   3    1 2       3   2    
      -x x  + x x  + 1, -x x  + ---x x  + -x x  + --x x x  + -x x x  +
      2 1 2    1 4      6 1 2    70 1 2   7 1 2   10 1 2 3   2 1 2 3  
      -----------------------------------------------------------------------
      5 2       6   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      3 1 2 4   7 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     10                   4                      16 2  
o16 = (map(R,R,{-x  + --x  + x , x , 4x  + -x  + x , x }), ideal (--x  +
                9 1    3 2    4   1    1   3 2    3   2            9 1  
      -----------------------------------------------------------------------
      10                 28 3     388 2 2   40   3   7 2       10   2    
      --x x  + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + --x x x  +
       3 1 2    1 4       9 1 2    27 1 2    9 1 2   9 1 2 3    3 1 2 3  
      -----------------------------------------------------------------------
        2       4   2
      4x x x  + -x x x  + x x x x  + 1), {x , x })
        1 2 4   3 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

                                                                  2          
o19 = (map(R,R,{6x  + 3x  + x , x , 3x  + x  + x , x }), ideal (7x  + 3x x  +
                  1     2    4   1    1    2    3   2             1     1 2  
      -----------------------------------------------------------------------
                   3        2 2       3     2           2       2      
      x x  + 1, 18x x  + 15x x  + 3x x  + 6x x x  + 3x x x  + 3x x x  +
       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 :