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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

                1     5             1     1                      11 2   5    
o3 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
               10 1   9 2    4   1  5 1   5 2    3   2           10 1   9 1 2
     ------------------------------------------------------------------------
                  1 3      59 2 2   1   3    1 2       5   2     1 2      
     + x x  + 1, --x x  + ---x x  + -x x  + --x x x  + -x x x  + -x x x  +
        1 4      50 1 2   450 1 2   9 1 2   10 1 2 3   9 1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
     1   2
     -x x x  + x x x x  + 1), {x , x })
     5 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

                     5             7                    2                    
o6 = (map(R,R,{5x  + -x  + x , x , -x  + x  + x , 4x  + -x  + x , x }), ideal
                 1   4 2    5   1  9 1    2    4    1   9 2    3   2         
     ------------------------------------------------------------------------
        2   5               3      3     375 2 2      2       375   3  
     (5x  + -x x  + x x  - x , 125x x  + ---x x  + 75x x x  + ---x x  +
        1   4 1 2    1 5    2      1 2    4  1 2      1 2 5    16 1 2  
     ------------------------------------------------------------------------
     75   2            2   125 4   75 3     15 2 2      3
     --x x x  + 15x x x  + ---x  + --x x  + --x x  + x x ), {x , x , x })
      2 1 2 5      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} | 5120x_1x_2x_5^6-240000x_2^9x_5-15625x_2^9+96000x_2^8x_5^2+
     {-9}  | 2500x_1x_2^2x_5^3-15360x_1x_2x_5^5+2000x_1x_2x_5^4+720000x
     {-9}  | 7812500x_1x_2^3+48000000x_1x_2^2x_5^2+12500000x_1x_2^2x_5+
     {-3}  | 20x_1^2+5x_1x_2+4x_1x_5-4x_2^3                            
     ------------------------------------------------------------------------
                                                                  
     12500x_2^8x_5-25600x_2^7x_5^3-10000x_2^7x_5^2+8000x_2^6x_5^3-
     _2^9-288000x_2^8x_5-12500x_2^8+76800x_2^7x_5^2+20000x_2^7x_5-
     1509949440x_1x_2x_5^5-98304000x_1x_2x_5^4+25600000x_1x_2x_5^3
                                                                  
     ------------------------------------------------------------------------
                                                                             
     6400x_2^5x_5^4+5120x_2^4x_5^5+1280x_2^2x_5^6+1024x_2x_5^7               
     24000x_2^6x_5^2+19200x_2^5x_5^3-15360x_2^4x_5^4+2000x_2^4x_5^3+625x_2^3x
     +5000000x_1x_2x_5^2-70778880000x_2^9+28311552000x_2^8x_5+1843200000x_2^8
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     _5^3-3840x_2^2x_5^5+1000x_2^2x_5^4-3072x_2x_5^6+400x_2x_5^5             
     -7549747200x_2^7x_5^2-2457600000x_2^7x_5+64000000x_2^7+2359296000x_2^6x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5^2-153600000x_2^6x_5-20000000x_2^6-1887436800x_2^5x_5^3+122880000x_2^5x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5^2+16000000x_2^5x_5+6250000x_2^5+1509949440x_2^4x_5^4-98304000x_2^4x_5
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     ^3+25600000x_2^4x_5^2+5000000x_2^4x_5+1953125x_2^4+12000000x_2^3x_5^2+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     4687500x_2^3x_5+377487360x_2^2x_5^5-24576000x_2^2x_5^4+16000000x_2^2x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     3+3750000x_2^2x_5^2+301989888x_2x_5^6-19660800x_2x_5^5+5120000x_2x_5^4+
                                                                            
     ------------------------------------------------------------------------
                     |
                     |
                     |
     1000000x_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

                8     10             7     9                      15 2  
o13 = (map(R,R,{-x  + --x  + x , x , -x  + -x  + x , x }), ideal (--x  +
                7 1    3 2    4   1  2 1   5 2    3   2            7 1  
      -----------------------------------------------------------------------
      10                   3     1441 2 2       3   8 2       10   2    
      --x x  + x x  + 1, 4x x  + ----x x  + 6x x  + -x x x  + --x x x  +
       3 1 2    1 4        1 2    105 1 2     1 2   7 1 2 3    3 1 2 3  
      -----------------------------------------------------------------------
      7 2       9   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      2 1 2 4   5 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

                3                  1     10                      7 2         
o16 = (map(R,R,{-x  + x  + x , x , -x  + --x  + x , x }), ideal (-x  + x x  +
                4 1    2    4   1  3 1    7 2    3   2           4 1    1 2  
      -----------------------------------------------------------------------
                1 3     59 2 2   10   3   3 2          2     1 2      
      x x  + 1, -x x  + --x x  + --x x  + -x x x  + x x x  + -x x x  +
       1 4      4 1 2   42 1 2    7 1 2   4 1 2 3    1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      10   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 , - x  + 2x  + x , x }), ideal (- x  +
                    1    2    4   1     1     2    3   2              1  
      -----------------------------------------------------------------------
                         3       2 2       3     2          2      2      
      x x  + x x  + 1, 2x x  - 5x x  + 2x x  - 2x x x  + x x x  - x x x  +
       1 2    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 :