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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

               4     10             1     1                      7 2   10    
o3 = (map(R,R,{-x  + --x  + x , x , -x  + -x  + x , x }), ideal (-x  + --x x 
               3 1    7 2    4   1  5 1   7 2    3   2           3 1    7 1 2
     ------------------------------------------------------------------------
                  4 3     10 2 2   10   3   4 2       10   2     1 2      
     + x x  + 1, --x x  + --x x  + --x x  + -x x x  + --x x x  + -x x x  +
        1 4      15 1 2   21 1 2   49 1 2   3 1 2 3    7 1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
     1   2
     -x x x  + x x x x  + 1), {x , x })
     7 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

                    2             3     3               1                    
o6 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , 3x  + -x  + x , x }), ideal
                1   3 2    5   1  4 1   2 2    4    1   5 2    3   2         
     ------------------------------------------------------------------------
       2   2               3   3       2 2     2       4   3       2    
     (x  + -x x  + x x  - x , x x  + 2x x  + 3x x x  + -x x  + 4x x x  +
       1   3 1 2    1 5    2   1 2     1 2     1 2 5   3 1 2     1 2 5  
     ------------------------------------------------------------------------
           2    8 4   4 3       2 2      3
     3x x x  + --x  + -x x  + 2x x  + x x ), {x , x , x })
       1 2 5   27 2   3 2 5     2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                      
     {-10} | 243x_1x_2x_5^6-648x_2^9x_5-32x_2^9+486x_2^8x_5^2+48x_2^8x_5-
     {-9}  | 48x_1x_2^2x_5^3-729x_1x_2x_5^5+72x_1x_2x_5^4+1944x_2^9-1458x
     {-9}  | 3072x_1x_2^3+46656x_1x_2^2x_5^2+9216x_1x_2^2x_5+4782969x_1x_
     {-3}  | 3x_1^2+2x_1x_2+3x_1x_5-3x_2^3                               
     ------------------------------------------------------------------------
                                                                          
     243x_2^7x_5^3-72x_2^7x_5^2+108x_2^6x_5^3-162x_2^5x_5^4+243x_2^4x_5^5+
     _2^8x_5-48x_2^8+729x_2^7x_5^2+144x_2^7x_5-324x_2^6x_5^2+486x_2^5x_5^3
     2x_5^5-236196x_1x_2x_5^4+46656x_1x_2x_5^3+6912x_1x_2x_5^2-12754584x_2
                                                                          
     ------------------------------------------------------------------------
                                                                             
     162x_2^2x_5^6+243x_2x_5^7                                               
     -729x_2^4x_5^4+72x_2^4x_5^3+32x_2^3x_5^3-486x_2^2x_5^5+96x_2^2x_5^4-729x
     ^9+9565938x_2^8x_5+472392x_2^8-4782969x_2^7x_5^2-1180980x_2^7x_5+23328x_
                                                                             
     ------------------------------------------------------------------------
                                                                       
                                                                       
     _2x_5^6+72x_2x_5^5                                                
     2^7+2125764x_2^6x_5^2-104976x_2^6x_5-10368x_2^6-3188646x_2^5x_5^3+
                                                                       
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     157464x_2^5x_5^2+15552x_2^5x_5+4608x_2^5+4782969x_2^4x_5^4-236196x_2^4x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5^3+46656x_2^4x_5^2+6912x_2^4x_5+2048x_2^4+31104x_2^3x_5^2+9216x_2^3x_5+
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     3188646x_2^2x_5^5-157464x_2^2x_5^4+77760x_2^2x_5^3+13824x_2^2x_5^2+
                                                                        
     ------------------------------------------------------------------------
                                                               |
                                                               |
                                                               |
     4782969x_2x_5^6-236196x_2x_5^5+46656x_2x_5^4+6912x_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             2     5                      7 2  
o13 = (map(R,R,{-x  + --x  + x , x , -x  + -x  + x , x }), ideal (-x  +
                5 1   10 2    4   1  5 1   3 2    3   2           5 1  
      -----------------------------------------------------------------------
       1                  4 3     53 2 2   1   3   2 2        1   2    
      --x x  + x x  + 1, --x x  + --x x  + -x x  + -x x x  + --x x x  +
      10 1 2    1 4      25 1 2   75 1 2   6 1 2   5 1 2 3   10 1 2 3  
      -----------------------------------------------------------------------
      2 2       5   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      5 1 2 4   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

                3     7             4     8                      5 2   7    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                2 1   5 2    4   1  3 1   3 2    3   2           2 1   5 1 2
      -----------------------------------------------------------------------
                    3     88 2 2   56   3   3 2       7   2     4 2      
      + x x  + 1, 2x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4        1 2   15 1 2   15 1 2   2 1 2 3   5 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      8   2
      -x x x  + x x x x  + 1), {x , x })
      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

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