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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

               2                  6     6                      7 2         
o3 = (map(R,R,{-x  + x  + x , x , -x  + -x  + x , x }), ideal (-x  + x x  +
               5 1    2    4   1  5 1   5 2    3   2           5 1    1 2  
     ------------------------------------------------------------------------
               12 3     42 2 2   6   3   2 2          2     6 2       6   2
     x x  + 1, --x x  + --x x  + -x x  + -x x x  + x x x  + -x x x  + -x x x 
      1 4      25 1 2   25 1 2   5 1 2   5 1 2 3    1 2 3   5 1 2 4   5 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             3               5                          
o6 = (map(R,R,{x  + -x  + x , x , -x  + 2x  + x , -x  + 3x  + x , x }), ideal
                1   2 2    5   1  7 1     2    4  9 1     2    3   2         
     ------------------------------------------------------------------------
       2   7               3   3     21 2 2     2       147   3        2    
     (x  + -x x  + x x  - x , x x  + --x x  + 3x x x  + ---x x  + 21x x x  +
       1   2 1 2    1 5    2   1 2    2 1 2     1 2 5    4  1 2      1 2 5  
     ------------------------------------------------------------------------
           2   343 4   147 3     21 2 2      3
     3x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
       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} | 32x_1x_2x_5^6-2352x_2^9x_5-16807x_2^9+336x_2^8x_5^2+4802x_2^8x_
     {-9}  | 4802x_1x_2^2x_5^3-96x_1x_2x_5^5+1372x_1x_2x_5^4+7056x_2^9-1008x
     {-9}  | 3954653486x_1x_2^3+79060128x_1x_2^2x_5^2+2259801992x_1x_2^2x_5+
     {-3}  | 2x_1^2+7x_1x_2+2x_1x_5-2x_2^3                                  
     ------------------------------------------------------------------------
                                                                            
     5-32x_2^7x_5^3-1372x_2^7x_5^2+392x_2^6x_5^3-112x_2^5x_5^4+32x_2^4x_5^5+
     _2^8x_5-4802x_2^8+96x_2^7x_5^2+2744x_2^7x_5-1176x_2^6x_5^2+336x_2^5x_5^
     73728x_1x_2x_5^5-526848x_1x_2x_5^4+15059072x_1x_2x_5^3+322828856x_1x_2x
                                                                            
     ------------------------------------------------------------------------
                                                                             
     112x_2^2x_5^6+32x_2x_5^7                                                
     3-96x_2^4x_5^4+1372x_2^4x_5^3+16807x_2^3x_5^3-336x_2^2x_5^5+9604x_2^2x_5
     _5^2-5419008x_2^9+774144x_2^8x_5+5531904x_2^8-73728x_2^7x_5^2-2634240x_2
                                                                             
     ------------------------------------------------------------------------
                                                                       
                                                                       
     ^4-96x_2x_5^6+1372x_2x_5^5                                        
     ^7x_5+7529536x_2^7+903168x_2^6x_5^2-6453888x_2^6x_5-92236816x_2^6-
                                                                       
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     258048x_2^5x_5^3+1843968x_2^5x_5^2+26353376x_2^5x_5+1129900996x_2^5+
                                                                         
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     73728x_2^4x_5^4-526848x_2^4x_5^3+15059072x_2^4x_5^2+322828856x_2^4x_5+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     13841287201x_2^4+276710448x_2^3x_5^2+11863960458x_2^3x_5+258048x_2^2x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5-1843968x_2^2x_5^4+131766880x_2^2x_5^3+3389702988x_2^2x_5^2+73728x_2x_5
                                                                             
     ------------------------------------------------------------------------
                                                          |
                                                          |
                                                          |
     ^6-526848x_2x_5^5+15059072x_2x_5^4+322828856x_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

                7                  1     8                      11 2         
o13 = (map(R,R,{-x  + x  + x , x , -x  + -x  + x , x }), ideal (--x  + x x  +
                4 1    2    4   1  5 1   7 2    3   2            4 1    1 2  
      -----------------------------------------------------------------------
                 7 3     11 2 2   8   3   7 2          2     1 2      
      x x  + 1, --x x  + --x x  + -x x  + -x x x  + x x x  + -x x x  +
       1 4      20 1 2    5 1 2   7 1 2   4 1 2 3    1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      8   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

                2     3                   5                      11 2   3    
o16 = (map(R,R,{-x  + -x  + x , x , 4x  + -x  + x , x }), ideal (--x  + -x x 
                9 1   4 2    4   1    1   4 2    3   2            9 1   4 1 2
      -----------------------------------------------------------------------
                  8 3     59 2 2   15   3   2 2       3   2       2      
      + x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + 4x x x  +
         1 4      9 1 2   18 1 2   16 1 2   9 1 2 3   4 1 2 3     1 2 4  
      -----------------------------------------------------------------------
      5   2
      -x x x  + x x x x  + 1), {x , x })
      4 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 , - 2x  + 2x  + x , x }), ideal (x  - 2x x  +
                    2    4   1      1     2    3   2            1     1 2  
      -----------------------------------------------------------------------
                  2 2       3       2       2           2
      x x  + 1, 4x x  - 4x x  - 2x x x  - 2x x x  + 2x x x  + x x x x  + 1),
       1 4        1 2     1 2     1 2 3     1 2 4     1 2 4    1 2 3 4      
      -----------------------------------------------------------------------
      {x , x })
        4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :