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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

               6     2                  1                      11 2   2      
o3 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (--x  + -x x  +
               5 1   5 2    4   1   1   4 2    3   2            5 1   5 1 2  
     ------------------------------------------------------------------------
               6 3      7 2 2    1   3   6 2       2   2      2       1   2
     x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + x x x  + -x x x 
      1 4      5 1 2   10 1 2   10 1 2   5 1 2 3   5 1 2 3    1 2 4   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

                     2                   3         7                    
o6 = (map(R,R,{5x  + -x  + x , x , x  + --x  + x , -x  + 3x  + x , x }),
                 1   5 2    5   1   1   10 2    4  4 1     2    3   2   
     ------------------------------------------------------------------------
              2   2               3      3        2 2      2       12   3  
     ideal (5x  + -x x  + x x  - x , 125x x  + 30x x  + 75x x x  + --x x  +
              1   5 1 2    1 5    2      1 2      1 2      1 2 5    5 1 2  
     ------------------------------------------------------------------------
          2            2    8  4   12 3     6 2 2      3
     12x x x  + 15x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
        1 2 5      1 2 5   125 2   25 2 5   5 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                             
     {-10} | 3125x_1x_2x_5^6-15000x_2^9x_5-32x_2^9+18750x_2^8x_5
     {-9}  | 400x_1x_2^2x_5^3-234375x_1x_2x_5^5+1000x_1x_2x_5^4+
     {-9}  | 25600x_1x_2^3+15000000x_1x_2^2x_5^2+128000x_1x_2^2x
     {-3}  | 25x_1^2+2x_1x_2+5x_1x_5-5x_2^3                     
     ------------------------------------------------------------------------
                                                                           
     ^2+80x_2^8x_5-15625x_2^7x_5^3-200x_2^7x_5^2+500x_2^6x_5^3-1250x_2^5x_5
     1125000x_2^9-1406250x_2^8x_5-2000x_2^8+1171875x_2^7x_5^2+10000x_2^7x_5
     _5+1373291015625x_1x_2x_5^5-2929687500x_1x_2x_5^4+25000000x_1x_2x_5^3+
                                                                           
     ------------------------------------------------------------------------
                                                                            
     ^4+3125x_2^4x_5^5+250x_2^2x_5^6+625x_2x_5^7                            
     -37500x_2^6x_5^2+93750x_2^5x_5^3-234375x_2^4x_5^4+1000x_2^4x_5^3+32x_2^
     160000x_1x_2x_5^2-6591796875000x_2^9+8239746093750x_2^8x_5+17578125000x
                                                                            
     ------------------------------------------------------------------------
                                                                    
                                                                    
     3x_5^3-18750x_2^2x_5^5+160x_2^2x_5^4-46875x_2x_5^6+200x_2x_5^5 
     _2^8-6866455078125x_2^7x_5^2-73242187500x_2^7x_5+62500000x_2^7+
                                                                    
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     219726562500x_2^6x_5^2-468750000x_2^6x_5-2000000x_2^6-549316406250x_2^5x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5^3+1171875000x_2^5x_5^2+5000000x_2^5x_5+64000x_2^5+1373291015625x_2^4x
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     _5^4-2929687500x_2^4x_5^3+25000000x_2^4x_5^2+160000x_2^4x_5+2048x_2^4+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1200000x_2^3x_5^2+15360x_2^3x_5+109863281250x_2^2x_5^5-234375000x_2^2x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^4+5000000x_2^2x_5^3+38400x_2^2x_5^2+274658203125x_2x_5^6-585937500x_2x_
                                                                             
     ------------------------------------------------------------------------
                                       |
                                       |
                                       |
     5^5+5000000x_2x_5^4+32000x_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             10     8                        2   4    
o13 = (map(R,R,{4x  + -x  + x , x , --x  + -x  + x , x }), ideal (5x  + -x x 
                  1   7 2    4   1   3 1   3 2    3   2             1   7 1 2
      -----------------------------------------------------------------------
                  40 3     88 2 2   32   3     2       4   2     10 2      
      + x x  + 1, --x x  + --x x  + --x x  + 4x x x  + -x x x  + --x x x  +
         1 4       3 1 2    7 1 2   21 1 2     1 2 3   7 1 2 3    3 1 2 4  
      -----------------------------------------------------------------------
      8   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

                      2                    1                        2   2    
o16 = (map(R,R,{5x  + -x  + x , x , 6x  + --x  + x , x }), ideal (6x  + -x x 
                  1   3 2    4   1    1   10 2    3   2             1   3 1 2
      -----------------------------------------------------------------------
                     3     9 2 2    1   3     2       2   2       2      
      + x x  + 1, 30x x  + -x x  + --x x  + 5x x x  + -x x x  + 6x x x  +
         1 4         1 2   2 1 2   15 1 2     1 2 3   3 1 2 3     1 2 4  
      -----------------------------------------------------------------------
       1   2
      --x x x  + x x x x  + 1), {x , x })
      10 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

                                                                        2  
o19 = (map(R,R,{- 4x  - x  + x , x , - 4x  - 6x  + x , x }), ideal (- 3x  -
                    1    2    4   1      1     2    3   2               1  
      -----------------------------------------------------------------------
                          3        2 2       3     2          2       2      
      x x  + x x  + 1, 16x x  + 28x x  + 6x x  - 4x 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
      6x 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 :