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

               6     3                  3                      13 2   3      
o3 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (--x  + -x x  +
               7 1   2 2    4   1   1   5 2    3   2            7 1   2 1 2  
     ------------------------------------------------------------------------
               6 3     141 2 2    9   3   6 2       3   2      2      
     x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + x x x  +
      1 4      7 1 2    70 1 2   10 1 2   7 1 2 3   2 1 2 3    1 2 4  
     ------------------------------------------------------------------------
     3   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

               1                         5         4     3              
o6 = (map(R,R,{-x  + 2x  + x , x , 2x  + -x  + x , -x  + -x  + x , x }),
               3 1     2    5   1    1   9 2    4  5 1   5 2    3   2   
     ------------------------------------------------------------------------
            1 2                   3   1 3     2 2 2   1 2           3  
     ideal (-x  + 2x x  + x x  - x , --x x  + -x x  + -x x x  + 4x x  +
            3 1     1 2    1 5    2  27 1 2   3 1 2   3 1 2 5     1 2  
     ------------------------------------------------------------------------
         2          2     4      3       2 2      3
     4x x x  + x x x  + 8x  + 12x x  + 6x x  + x x ), {x , x , x })
       1 2 5    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} | 3x_1x_2x_5^6-24x_2^9x_5-96x_2^9+6x_2^8x_5^2+48x_2^8x_5-x_2^
     {-9}  | 48x_1x_2^2x_5^3-3x_1x_2x_5^5+24x_1x_2x_5^4+24x_2^9-6x_2^8x_
     {-9}  | 9216x_1x_2^3+576x_1x_2^2x_5^2+9216x_1x_2^2x_5+3x_1x_2x_5^5-
     {-3}  | x_1^2+6x_1x_2+3x_1x_5-3x_2^3                               
     ------------------------------------------------------------------------
                                                                             
     7x_5^3-24x_2^7x_5^2+12x_2^6x_5^3-6x_2^5x_5^4+3x_2^4x_5^5+18x_2^2x_5^6+9x
     5-16x_2^8+x_2^7x_5^2+16x_2^7x_5-12x_2^6x_5^2+6x_2^5x_5^3-3x_2^4x_5^4+24x
     12x_1x_2x_5^4+192x_1x_2x_5^3+2304x_1x_2x_5^2-24x_2^9+6x_2^8x_5+24x_2^8-x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _2x_5^7                                                                 
     _2^4x_5^3+288x_2^3x_5^3-18x_2^2x_5^5+288x_2^2x_5^4-9x_2x_5^6+72x_2x_5^5 
     _2^7x_5^2-20x_2^7x_5+32x_2^7+12x_2^6x_5^2-48x_2^6x_5-384x_2^6-6x_2^5x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3+24x_2^5x_5^2+192x_2^5x_5+4608x_2^5+3x_2^4x_5^4-12x_2^4x_5^3+192x_2^4x_
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     5^2+2304x_2^4x_5+55296x_2^4+3456x_2^3x_5^2+82944x_2^3x_5+18x_2^2x_5^5-
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     72x_2^2x_5^4+2880x_2^2x_5^3+41472x_2^2x_5^2+9x_2x_5^6-36x_2x_5^5+576x_2x
                                                                             
     ------------------------------------------------------------------------
                       |
                       |
                       |
     _5^4+6912x_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

                      1             4     7                        2   1    
o13 = (map(R,R,{2x  + -x  + x , x , -x  + -x  + x , x }), ideal (3x  + -x x 
                  1   2 2    4   1  7 1   3 2    3   2             1   2 1 2
      -----------------------------------------------------------------------
                  8 3     104 2 2   7   3     2       1   2     4 2      
      + x x  + 1, -x x  + ---x x  + -x x  + 2x x x  + -x x x  + -x x x  +
         1 4      7 1 2    21 1 2   6 1 2     1 2 3   2 1 2 3   7 1 2 4  
      -----------------------------------------------------------------------
      7   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

                7     7              7                            16 2  
o16 = (map(R,R,{-x  + -x  + x , x , --x  + 7x  + x , x }), ideal (--x  +
                9 1   6 2    4   1  10 1     2    3   2            9 1  
      -----------------------------------------------------------------------
      7                 49 3     1127 2 2   49   3   7 2       7   2    
      -x x  + x x  + 1, --x x  + ----x x  + --x x  + -x x x  + -x x x  +
      6 1 2    1 4      90 1 2    180 1 2    6 1 2   9 1 2 3   6 1 2 3  
      -----------------------------------------------------------------------
       7 2           2
      --x x x  + 7x x x  + x x x x  + 1), {x , x })
      10 1 2 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
--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

                                                          2                 
o19 = (map(R,R,{- 4x  + x , x , - 2x  + x , x }), ideal (x  - 4x x  + x x  +
                    2    4   1      2    3   2            1     1 2    1 4  
      -----------------------------------------------------------------------
             3       2         2
      1, 8x x  - 4x x x  - 2x x x  + x x x x  + 1), {x , x })
           1 2     1 2 3     1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :