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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     1                                          8 2   1      
o3 = (map(R,R,{-x  + -x  + x , x , 2x  + 2x  + x , x }), ideal (-x  + -x x  +
               3 1   3 2    4   1    1     2    3   2           3 1   3 1 2  
     ------------------------------------------------------------------------
               10 3       2 2   2   3   5 2       1   2       2           2
     x x  + 1, --x x  + 4x x  + -x x  + -x x x  + -x x x  + 2x x x  + 2x x x 
      1 4       3 1 2     1 2   3 1 2   3 1 2 3   3 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     9                  7         1     2                    
o6 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , -x  + -x  + x , x }), ideal
               2 1   4 2    5   1   1   3 2    4  2 1   9 2    3   2         
     ------------------------------------------------------------------------
      1 2   9               3  1 3     27 2 2   3 2       243   3   27   2  
     (-x  + -x x  + x x  - x , -x x  + --x x  + -x x x  + ---x x  + --x x x 
      2 1   4 1 2    1 5    2  8 1 2   16 1 2   4 1 2 5    32 1 2    4 1 2 5
     ------------------------------------------------------------------------
       3     2   729 4   243 3     27 2 2      3
     + -x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
       2 1 2 5    64 2    16 2 5    4 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 1024x_1x_2x_5^6-15552x_2^9x_5-59049x_2^9+3456x_2^8x_5^2+26244x_2
     {-9}  | 4374x_1x_2^2x_5^3-256x_1x_2x_5^5+1944x_1x_2x_5^4+3888x_2^9-864x_
     {-9}  | 6973568802x_1x_2^3+408146688x_1x_2^2x_5^2+6198727824x_1x_2^2x_5+
     {-3}  | 2x_1^2+9x_1x_2+4x_1x_5-4x_2^3                                   
     ------------------------------------------------------------------------
                                                                            
     ^8x_5-512x_2^7x_5^3-11664x_2^7x_5^2+5184x_2^6x_5^3-2304x_2^5x_5^4+1024x
     2^8x_5-2187x_2^8+128x_2^7x_5^2+1944x_2^7x_5-1296x_2^6x_5^2+576x_2^5x_5^
     2097152x_1x_2x_5^5-7962624x_1x_2x_5^4+120932352x_1x_2x_5^3+1377495072x_
                                                                            
     ------------------------------------------------------------------------
                                                                            
     _2^4x_5^5+4608x_2^2x_5^6+2048x_2x_5^7                                  
     3-256x_2^4x_5^4+1944x_2^4x_5^3+19683x_2^3x_5^3-1152x_2^2x_5^5+17496x_2^
     1x_2x_5^2-31850496x_2^9+7077888x_2^8x_5+26873856x_2^8-1048576x_2^7x_5^2
                                                                            
     ------------------------------------------------------------------------
                                                                         
                                                                         
     2x_5^4-512x_2x_5^6+3888x_2x_5^5                                     
     -19906560x_2^7x_5+30233088x_2^7+10616832x_2^6x_5^2-40310784x_2^6x_5-
                                                                         
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     306110016x_2^6-4718592x_2^5x_5^3+17915904x_2^5x_5^2+136048896x_2^5x_5+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3099363912x_2^5+2097152x_2^4x_5^4-7962624x_2^4x_5^3+120932352x_2^4x_5^2+
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1377495072x_2^4x_5+31381059609x_2^4+1836660096x_2^3x_5^2+41841412812x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     3x_5+9437184x_2^2x_5^5-35831808x_2^2x_5^4+1360488960x_2^2x_5^3+
                                                                    
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     18596183472x_2^2x_5^2+4194304x_2x_5^6-15925248x_2x_5^5+241864704x_2x_5^4
                                                                             
     ------------------------------------------------------------------------
                         |
                         |
                         |
     +2754990144x_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

                3     1             4     4                      7 2   1    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                4 1   5 2    4   1  3 1   3 2    3   2           4 1   5 1 2
      -----------------------------------------------------------------------
                   3     19 2 2    4   3   3 2       1   2     4 2      
      + x x  + 1, x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4       1 2   15 1 2   15 1 2   4 1 2 3   5 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      4   2
      -x x x  + x x x x  + 1), {x , x })
      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

                1                   5     4                      5 2        
o16 = (map(R,R,{-x  + 5x  + x , x , -x  + -x  + x , x }), ideal (-x  + 5x x 
                4 1     2    4   1  7 1   5 2    3   2           4 1     1 2
      -----------------------------------------------------------------------
                   5 3     132 2 2       3   1 2           2     5 2      
      + x x  + 1, --x x  + ---x x  + 4x x  + -x x x  + 5x x x  + -x x x  +
         1 4      28 1 2    35 1 2     1 2   4 1 2 3     1 2 3   7 1 2 4  
      -----------------------------------------------------------------------
      4   2
      -x x x  + x x x x  + 1), {x , x })
      5 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,{2x  - 2x  + x , x , 2x  - x  + x , x }), ideal (3x  - 2x x  +
                  1     2    4   1    1    2    3   2             1     1 2  
      -----------------------------------------------------------------------
                  3       2 2       3     2           2       2          2
      x x  + 1, 4x x  - 6x x  + 2x x  + 2x x x  - 2x x x  + 2x x x  - x x x 
       1 4        1 2     1 2     1 2     1 2 3     1 2 3     1 2 4    1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :