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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

               10     9             1     5                      17 2   9    
o3 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                7 1   7 2    4   1  2 1   3 2    3   2            7 1   7 1 2
     ------------------------------------------------------------------------
                 5 3     127 2 2   15   3   10 2       9   2     1 2      
     + x x  + 1, -x x  + ---x x  + --x x  + --x x x  + -x x x  + -x x x  +
        1 4      7 1 2    42 1 2    7 1 2    7 1 2 3   7 1 2 3   2 1 2 4  
     ------------------------------------------------------------------------
     5   2
     -x x x  + x x x x  + 1), {x , x })
     3 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

               3                  1               3     7                    
o6 = (map(R,R,{-x  + x  + x , x , -x  + 9x  + x , -x  + -x  + x , x }), ideal
               8 1    2    5   1  5 1     2    4  2 1   9 2    3   2         
     ------------------------------------------------------------------------
      3 2                  3   27 3     27 2 2   27 2       9   3   9   2    
     (-x  + x x  + x x  - x , ---x x  + --x x  + --x x x  + -x x  + -x x x  +
      8 1    1 2    1 5    2  512 1 2   64 1 2   64 1 2 5   8 1 2   4 1 2 5  
     ------------------------------------------------------------------------
     9     2    4     3       2 2      3
     -x x x  + x  + 3x x  + 3x x  + x x ), {x , x , x })
     8 1 2 5    2     2 5     2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                        
     {-10} | 24x_1x_2x_5^6-54x_2^9x_5-24x_2^9+27x_2^8x_5^2+24x_2^8x_5-9x_2^
     {-9}  | 192x_1x_2^2x_5^3-216x_1x_2x_5^5+192x_1x_2x_5^4+486x_2^9-243x_2
     {-9}  | 6144x_1x_2^3+6912x_1x_2^2x_5^2+12288x_1x_2^2x_5+5832x_1x_2x_5^
     {-3}  | 3x_1^2+8x_1x_2+8x_1x_5-8x_2^3                                 
     ------------------------------------------------------------------------
                                                                             
     7x_5^3-24x_2^7x_5^2+24x_2^6x_5^3-24x_2^5x_5^4+24x_2^4x_5^5+64x_2^2x_5^6+
     ^8x_5-72x_2^8+81x_2^7x_5^2+144x_2^7x_5-216x_2^6x_5^2+216x_2^5x_5^3-216x_
     5-2592x_1x_2x_5^4+4608x_1x_2x_5^3+6144x_1x_2x_5^2-13122x_2^9+6561x_2^8x_
                                                                             
     ------------------------------------------------------------------------
                                                                            
     64x_2x_5^7                                                             
     2^4x_5^4+192x_2^4x_5^3+512x_2^3x_5^3-576x_2^2x_5^5+1024x_2^2x_5^4-576x_
     5+2916x_2^8-2187x_2^7x_5^2-4860x_2^7x_5+864x_2^7+5832x_2^6x_5^2-2592x_2
                                                                            
     ------------------------------------------------------------------------
                                                                          
                                                                          
     2x_5^6+512x_2x_5^5                                                   
     ^6x_5-2304x_2^6-5832x_2^5x_5^3+2592x_2^5x_5^2+2304x_2^5x_5+6144x_2^5+
                                                                          
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     5832x_2^4x_5^4-2592x_2^4x_5^3+4608x_2^4x_5^2+6144x_2^4x_5+16384x_2^4+
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     18432x_2^3x_5^2+49152x_2^3x_5+15552x_2^2x_5^5-6912x_2^2x_5^4+30720x_2^2x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5^3+49152x_2^2x_5^2+15552x_2x_5^6-6912x_2x_5^5+12288x_2x_5^4+16384x_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     6             2     8                      9 2   6    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                2 1   5 2    4   1  3 1   7 2    3   2           2 1   5 1 2
      -----------------------------------------------------------------------
                  7 3     24 2 2   48   3   7 2       6   2     2 2      
      + x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      3 1 2    5 1 2   35 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 })
      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

                     9             5     10                        2   9    
o16 = (map(R,R,{x  + -x  + x , x , -x  + --x  + x , x }), ideal (2x  + -x x 
                 1   8 2    4   1  4 1    3 2    3   2             1   8 1 2
      -----------------------------------------------------------------------
                  5 3     455 2 2   15   3    2       9   2     5 2      
      + x x  + 1, -x x  + ---x x  + --x x  + x x x  + -x x x  + -x x x  +
         1 4      4 1 2    96 1 2    4 1 2    1 2 3   8 1 2 3   4 1 2 4  
      -----------------------------------------------------------------------
      10   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
--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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                        2  
o19 = (map(R,R,{- 3x  + x  + x , x , - 4x  + 4x  + x , x }), ideal (- 2x  +
                    1    2    4   1      1     2    3   2               1  
      -----------------------------------------------------------------------
                          3        2 2       3     2          2       2      
      x x  + x x  + 1, 12x x  - 16x x  + 4x x  - 3x x x  + x x x  - 4x x x  +
       1 2    1 4         1 2      1 2     1 2     1 2 3    1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      4x 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 :