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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                              2          
o3 = (map(R,R,{x  + 4x  + x , x , -x  + 3x  + x , x }), ideal (2x  + 4x x  +
                1     2    4   1  9 1     2    3   2             1     1 2  
     ------------------------------------------------------------------------
               7 3     55 2 2        3    2           2     7 2           2
     x x  + 1, -x x  + --x x  + 12x x  + x x x  + 4x x x  + -x x x  + 3x x x 
      1 4      9 1 2    9 1 2      1 2    1 2 3     1 2 3   9 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

               7     1             2                    2                    
o6 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , 5x  + -x  + x , x }), ideal
               4 1   2 2    5   1  3 1    2    4    1   9 2    3   2         
     ------------------------------------------------------------------------
      7 2   1               3  343 3     147 2 2   147 2       21   3  
     (-x  + -x x  + x x  - x , ---x x  + ---x x  + ---x x x  + --x x  +
      4 1   2 1 2    1 5    2   64 1 2    32 1 2    16 1 2 5   16 1 2  
     ------------------------------------------------------------------------
     21   2     21     2   1 4   3 3     3 2 2      3
     --x x x  + --x x x  + -x  + -x x  + -x x  + x x ), {x , x , x })
      4 1 2 5    4 1 2 5   8 2   4 2 5   2 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                  
     {-10} | 224x_1x_2x_5^6-588x_2^9x_5-7x_2^9+588x_2^8x_5^2+14x_2^8x
     {-9}  | 28x_1x_2^2x_5^3-2352x_1x_2x_5^5+56x_1x_2x_5^4+6174x_2^9-
     {-9}  | 7x_1x_2^3+588x_1x_2^2x_5^2+28x_1x_2^2x_5+1382976x_1x_2x_
     {-3}  | 7x_1^2+2x_1x_2+4x_1x_5-4x_2^3                           
     ------------------------------------------------------------------------
                                                                   
     _5-392x_2^7x_5^3-28x_2^7x_5^2+56x_2^6x_5^3-112x_2^5x_5^4+224x_
     6174x_2^8x_5-49x_2^8+4116x_2^7x_5^2+196x_2^7x_5-588x_2^6x_5^2+
     5^5-16464x_1x_2x_5^4+784x_1x_2x_5^3+28x_1x_2x_5^2-3630312x_2^9
                                                                   
     ------------------------------------------------------------------------
                                                                          
     2^4x_5^5+64x_2^2x_5^6+128x_2x_5^7                                    
     1176x_2^5x_5^3-2352x_2^4x_5^4+56x_2^4x_5^3+8x_2^3x_5^3-672x_2^2x_5^5+
     +3630312x_2^8x_5+43218x_2^8-2420208x_2^7x_5^2-144060x_2^7x_5+686x_2^7
                                                                          
     ------------------------------------------------------------------------
                                                                            
                                                                            
     32x_2^2x_5^4-1344x_2x_5^6+32x_2x_5^5                                   
     +345744x_2^6x_5^2-4116x_2^6x_5-98x_2^6-691488x_2^5x_5^3+8232x_2^5x_5^2+
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     196x_2^5x_5+14x_2^5+1382976x_2^4x_5^4-16464x_2^4x_5^3+784x_2^4x_5^2+28x_
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     2^4x_5+2x_2^4+168x_2^3x_5^2+12x_2^3x_5+395136x_2^2x_5^5-4704x_2^2x_5^4+
                                                                            
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     560x_2^2x_5^3+24x_2^2x_5^2+790272x_2x_5^6-9408x_2x_5^5+448x_2x_5^4+16x_
                                                                            
     ------------------------------------------------------------------------
            |
            |
            |
     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

                2     1             3     10                      9 2   1    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (-x  + -x x 
                7 1   9 2    4   1  2 1    7 2    3   2           7 1   9 1 2
      -----------------------------------------------------------------------
                  3 3     169 2 2   10   3   2 2       1   2     3 2      
      + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      7 1 2   294 1 2   63 1 2   7 1 2 3   9 1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      10   2
      --x x x  + x x x x  + 1), {x , x })
       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

                1     5                   5                      8 2   5    
o16 = (map(R,R,{-x  + -x  + x , x , 6x  + -x  + x , x }), ideal (-x  + -x x 
                7 1   2 2    4   1    1   2 2    3   2           7 1   2 1 2
      -----------------------------------------------------------------------
                  6 3     215 2 2   25   3   1 2       5   2       2      
      + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + 6x x x  +
         1 4      7 1 2    14 1 2    4 1 2   7 1 2 3   2 1 2 3     1 2 4  
      -----------------------------------------------------------------------
      5   2
      -x x x  + x x x x  + 1), {x , x })
      2 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  + x  + x , x , - 4x  - 6x  + x , x }), ideal (- 3x  +
                    1    2    4   1      1     2    3   2               1  
      -----------------------------------------------------------------------
                          3        2 2       3     2          2       2      
      x x  + x x  + 1, 16x x  + 20x x  - 6x x  - 4x x x  + x x x  - 4x x x  -
       1 2    1 4         1 2      1 2     1 2     1 2 3    1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      6x 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 :