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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

               9                  1                             17 2         
o3 = (map(R,R,{-x  + x  + x , x , -x  + 10x  + x , x }), ideal (--x  + x x  +
               8 1    2    4   1  2 1      2    3   2            8 1    1 2  
     ------------------------------------------------------------------------
                9 3     47 2 2        3   9 2          2     1 2      
     x x  + 1, --x x  + --x x  + 10x x  + -x x x  + x x x  + -x x x  +
      1 4      16 1 2    4 1 2      1 2   8 1 2 3    1 2 3   2 1 2 4  
     ------------------------------------------------------------------------
          2
     10x 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

               2     4             1      9         2     7              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , -x  + -x  + x , x }),
               3 1   3 2    5   1  8 1   10 2    4  9 1   4 2    3   2   
     ------------------------------------------------------------------------
            2 2   4               3   8 3     16 2 2   4 2       32   3  
     ideal (-x  + -x x  + x x  - x , --x x  + --x x  + -x x x  + --x x  +
            3 1   3 1 2    1 5    2  27 1 2    9 1 2   3 1 2 5    9 1 2  
     ------------------------------------------------------------------------
     16   2           2   64 4   16 3       2 2      3
     --x x x  + 2x x x  + --x  + --x x  + 4x x  + x x ), {x , x , x })
      3 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} | 486x_1x_2x_5^6-3456x_2^9x_5-2048x_2^9+1296x_2^8x_5^2+1536x_2^8x_
     {-9}  | 768x_1x_2^2x_5^3-486x_1x_2x_5^5+576x_1x_2x_5^4+3456x_2^9-1296x_2
     {-9}  | 524288x_1x_2^3+331776x_1x_2^2x_5^2+786432x_1x_2^2x_5+118098x_1x_
     {-3}  | 2x_1^2+4x_1x_2+3x_1x_5-3x_2^3                                   
     ------------------------------------------------------------------------
                                                                          
     5-324x_2^7x_5^3-1152x_2^7x_5^2+864x_2^6x_5^3-648x_2^5x_5^4+486x_2^4x_
     ^8x_5-512x_2^8+324x_2^7x_5^2+768x_2^7x_5-864x_2^6x_5^2+648x_2^5x_5^3-
     2x_5^5-69984x_1x_2x_5^4+165888x_1x_2x_5^3+294912x_1x_2x_5^2-839808x_2
                                                                          
     ------------------------------------------------------------------------
                                                                             
     5^5+972x_2^2x_5^6+729x_2x_5^7                                           
     486x_2^4x_5^4+576x_2^4x_5^3+1536x_2^3x_5^3-972x_2^2x_5^5+2304x_2^2x_5^4-
     ^9+314928x_2^8x_5+186624x_2^8-78732x_2^7x_5^2-233280x_2^7x_5+55296x_2^7+
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     729x_2x_5^6+864x_2x_5^5                                                 
     209952x_2^6x_5^2-124416x_2^6x_5-147456x_2^6-157464x_2^5x_5^3+93312x_2^5x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5^2+110592x_2^5x_5+393216x_2^5+118098x_2^4x_5^4-69984x_2^4x_5^3+165888x
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     _2^4x_5^2+294912x_2^4x_5+1048576x_2^4+663552x_2^3x_5^2+2359296x_2^3x_5+
                                                                            
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     236196x_2^2x_5^5-139968x_2^2x_5^4+829440x_2^2x_5^3+1769472x_2^2x_5^2+
                                                                          
     ------------------------------------------------------------------------
                                                                 |
                                                                 |
                                                                 |
     177147x_2x_5^6-104976x_2x_5^5+248832x_2x_5^4+442368x_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

                      4             5     2                        2   4    
o13 = (map(R,R,{3x  + -x  + x , x , -x  + -x  + x , x }), ideal (4x  + -x x 
                  1   3 2    4   1  2 1   3 2    3   2             1   3 1 2
      -----------------------------------------------------------------------
                  15 3     16 2 2   8   3     2       4   2     5 2      
      + x x  + 1, --x x  + --x x  + -x x  + 3x x x  + -x x x  + -x x x  +
         1 4       2 1 2    3 1 2   9 1 2     1 2 3   3 1 2 3   2 1 2 4  
      -----------------------------------------------------------------------
      2   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

                                         4                        2          
o16 = (map(R,R,{x  + 6x  + x , x , 8x  + -x  + x , x }), ideal (2x  + 6x x  +
                 1     2    4   1    1   7 2    3   2             1     1 2  
      -----------------------------------------------------------------------
                  3     340 2 2   24   3    2           2       2      
      x x  + 1, 8x x  + ---x x  + --x x  + x x x  + 6x x x  + 8x x x  +
       1 4        1 2    7  1 2    7 1 2    1 2 3     1 2 3     1 2 4  
      -----------------------------------------------------------------------
      4   2
      -x x x  + x x x x  + 1), {x , x })
      7 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  - x  + x , x , - 2x  - 2x  + x , x }), ideal (3x  - x x 
                  1    2    4   1      1     2    3   2             1    1 2
      -----------------------------------------------------------------------
                      3       2 2       3     2          2       2      
      + x x  + 1, - 4x x  - 2x x  + 2x x  + 2x x x  - x x x  - 2x x x  -
         1 4          1 2     1 2     1 2     1 2 3    1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      2x 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 :