next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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

               3     5             3     1                      5 2   5      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x  +
               2 1   3 2    4   1  5 1   8 2    3   2           2 1   3 1 2  
     ------------------------------------------------------------------------
                9 3     19 2 2    5   3   3 2       5   2     3 2      
     x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
      1 4      10 1 2   16 1 2   24 1 2   2 1 2 3   3 1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
     1   2
     -x x x  + x x x x  + 1), {x , x })
     8 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     2             5     5         4     4              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , -x  + -x  + x , x }),
               9 1   7 2    5   1  7 1   9 2    4  5 1   3 2    3   2   
     ------------------------------------------------------------------------
            5 2   2               3  125 3      50 2 2   25 2        20   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x  +
            9 1   7 1 2    1 5    2  729 1 2   189 1 2   27 1 2 5   147 1 2  
     ------------------------------------------------------------------------
     20   2     5     2    8  4   12 3     6 2 2      3
     --x x x  + -x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
     21 1 2 5   3 1 2 5   343 2   49 2 5   7 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                 
     {-10} | 756315x_1x_2x_5^6-205800x_2^9x_5-1440x_2^9+360150x_2^8x
     {-9}  | 15120x_1x_2^2x_5^3-3781575x_1x_2x_5^5+52920x_1x_2x_5^4+
     {-9}  | 26127360x_1x_2^3+6534561600x_1x_2^2x_5^2+182891520x_1x_
     {-3}  | 35x_1^2+18x_1x_2+63x_1x_5-63x_2^3                      
     ------------------------------------------------------------------------
                                                              
     _5^2+5040x_2^8x_5-420175x_2^7x_5^3-17640x_2^7x_5^2+61740x
     1029000x_2^9-1800750x_2^8x_5-8400x_2^8+2100875x_2^7x_5^2+
     2^2x_5+77857240505625x_1x_2x_5^5-544773694500x_1x_2x_5^4+
                                                              
     ------------------------------------------------------------------------
                                                                 
     _2^6x_5^3-216090x_2^5x_5^4+756315x_2^4x_5^5+388962x_2^2x_5^6
     58800x_2^7x_5-308700x_2^6x_5^2+1080450x_2^5x_5^3-3781575x_2^
     15247310400x_1x_2x_5^3+320060160x_1x_2x_5^2-21185643675000x_
                                                                 
     ------------------------------------------------------------------------
                                                                           
     +1361367x_2x_5^7                                                      
     4x_5^4+52920x_2^4x_5^3+7776x_2^3x_5^3-1944810x_2^2x_5^5+54432x_2^2x_5^
     2^9+37074876431250x_2^8x_5+259416045000x_2^8-43254022503125x_2^7x_5^2-
                                                                           
     ------------------------------------------------------------------------
                                                                   
                                                                   
     4-6806835x_2x_5^6+95256x_2x_5^5                               
     1513260262500x_2^7x_5+4235364000x_2^7+6355693102500x_2^6x_5^2-
                                                                   
     ------------------------------------------------------------------------
                                                                 
                                                                 
                                                                 
     44471322000x_2^6x_5-622339200x_2^6-22244925858750x_2^5x_5^3+
                                                                 
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     155649627000x_2^5x_5^2+2178187200x_2^5x_5+91445760x_2^5+77857240505625x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^4x_5^4-544773694500x_2^4x_5^3+15247310400x_2^4x_5^2+320060160x_2^4x_5+
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     13436928x_2^4+3360631680x_2^3x_5^2+141087744x_2^3x_5+40040866545750x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2x_5^5-280169328600x_2^2x_5^4+19603684800x_2^2x_5^3+493807104x_2^2x_5^2+
                                                                             
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     140143032910125x_2x_5^6-980592650100x_2x_5^5+27445158720x_2x_5^4+
                                                                      
     ------------------------------------------------------------------------
                       |
                       |
                       |
     576108288x_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

                10     9             10                            17 2  
o13 = (map(R,R,{--x  + -x  + x , x , --x  + 4x  + x , x }), ideal (--x  +
                 7 1   2 2    4   1   7 1     2    3   2            7 1  
      -----------------------------------------------------------------------
      9                 100 3     85 2 2        3   10 2       9   2    
      -x x  + x x  + 1, ---x x  + --x x  + 18x x  + --x x x  + -x x x  +
      2 1 2    1 4       49 1 2    7 1 2      1 2    7 1 2 3   2 1 2 3  
      -----------------------------------------------------------------------
      10 2           2
      --x x x  + 4x x x  + x x x x  + 1), {x , x })
       7 1 2 4     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     6                  1                      5 2   6      
o16 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (-x  + -x x  +
                3 1   5 2    4   1   1   9 2    3   2           3 1   5 1 2  
      -----------------------------------------------------------------------
                2 3     172 2 2    2   3   2 2       6   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   135 1 2   15 1 2   3 1 2 3   5 1 2 3    1 2 4  
      -----------------------------------------------------------------------
      1   2
      -x x x  + x x x x  + 1), {x , x })
      9 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,{- 3x  - 2x  + x , x , 4x  - 4x  + x , x }), ideal (- 2x  -
                    1     2    4   1    1     2    3   2               1  
      -----------------------------------------------------------------------
                             3       2 2       3     2           2    
      2x x  + x x  + 1, - 12x x  + 4x x  + 8x x  - 3x x x  - 2x x x  +
        1 2    1 4           1 2     1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      4x x x  - 4x x x  + x x x x  + 1), {x , x })
        1 2 4     1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :