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NoetherNormalization :: noetherNormalization

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

               5     7                  1                      9 2   7      
o3 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (-x  + -x x  +
               4 1   8 2    4   1   1   4 2    3   2           4 1   8 1 2  
     ------------------------------------------------------------------------
               5 3     19 2 2    7   3   5 2       7   2      2       1   2
     x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + x x x  + -x x x 
      1 4      4 1 2   16 1 2   32 1 2   4 1 2 3   8 1 2 3    1 2 4   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

                     8             8     3         8     5              
o6 = (map(R,R,{4x  + -x  + x , x , -x  + -x  + x , -x  + -x  + x , x }),
                 1   3 2    5   1  5 1   2 2    4  5 1   3 2    3   2   
     ------------------------------------------------------------------------
              2   8               3     3         2 2      2       256   3  
     ideal (4x  + -x x  + x x  - x , 64x x  + 128x x  + 48x x x  + ---x x  +
              1   3 1 2    1 5    2     1 2       1 2      1 2 5    3  1 2  
     ------------------------------------------------------------------------
          2            2   512 4   64 3       2 2      3
     64x x x  + 12x x x  + ---x  + --x x  + 8x x  + x x ), {x , x , x })
        1 2 5      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} | 972x_1x_2x_5^6-165888x_2^9x_5-131072x_2^9+31104x_2^
     {-9}  | 12288x_1x_2^2x_5^3-2916x_1x_2x_5^5+4608x_1x_2x_5^4+
     {-9}  | 805306368x_1x_2^3+191102976x_1x_2^2x_5^2+603979776x
     {-3}  | 12x_1^2+8x_1x_2+3x_1x_5-3x_2^3                     
     ------------------------------------------------------------------------
                                                                        
     8x_5^2+49152x_2^8x_5-3888x_2^7x_5^3-18432x_2^7x_5^2+6912x_2^6x_5^3-
     497664x_2^9-93312x_2^8x_5-49152x_2^8+11664x_2^7x_5^2+36864x_2^7x_5-
     _1x_2^2x_5+19131876x_1x_2x_5^5-15116544x_1x_2x_5^4+47775744x_1x_2x_
                                                                        
     ------------------------------------------------------------------------
                                                                             
     2592x_2^5x_5^4+972x_2^4x_5^5+648x_2^2x_5^6+243x_2x_5^7                  
     20736x_2^6x_5^2+7776x_2^5x_5^3-2916x_2^4x_5^4+4608x_2^4x_5^3+8192x_2^3x_
     5^3+113246208x_1x_2x_5^2-3265173504x_2^9+612220032x_2^8x_5+483729408x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     5^3-1944x_2^2x_5^5+6144x_2^2x_5^4-729x_2x_5^6+1152x_2x_5^5              
     8-76527504x_2^7x_5^2-302330880x_2^7x_5+95551488x_2^7+136048896x_2^6x_5^2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     -107495424x_2^6x_5-169869312x_2^6-51018336x_2^5x_5^3+40310784x_2^5x_5^2+
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     63700992x_2^5x_5+301989888x_2^5+19131876x_2^4x_5^4-15116544x_2^4x_5^3+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     47775744x_2^4x_5^2+113246208x_2^4x_5+536870912x_2^4+127401984x_2^3x_5^2+
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     603979776x_2^3x_5+12754584x_2^2x_5^5-10077696x_2^2x_5^4+79626240x_2^2x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^3+226492416x_2^2x_5^2+4782969x_2x_5^6-3779136x_2x_5^5+11943936x_2x_5^4+
                                                                             
     ------------------------------------------------------------------------
                      |
                      |
                      |
     28311552x_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

                10                   7                           19 2        
o13 = (map(R,R,{--x  + 2x  + x , x , -x  + x  + x , x }), ideal (--x  + 2x x 
                 9 1     2    4   1  9 1    2    3   2            9 1     1 2
      -----------------------------------------------------------------------
                  70 3     8 2 2       3   10 2           2     7 2      
      + x x  + 1, --x x  + -x x  + 2x x  + --x x x  + 2x x x  + -x x x  +
         1 4      81 1 2   3 1 2     1 2    9 1 2 3     1 2 3   9 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

                1     1             8     2                      6 2   1    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                5 1   6 2    4   1  7 1   9 2    3   2           5 1   6 1 2
      -----------------------------------------------------------------------
                   8 3      74 2 2    1   3   1 2       1   2     8 2      
      + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      35 1 2   315 1 2   27 1 2   5 1 2 3   6 1 2 3   7 1 2 4  
      -----------------------------------------------------------------------
      2   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
--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

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

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :