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     8             1                            5 2   8      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + 9x  + x , x }), ideal (-x  + -x x  +
               2 1   9 2    4   1  5 1     2    3   2           2 1   9 1 2  
     ------------------------------------------------------------------------
                3 3     1231 2 2       3   3 2       8   2     1 2      
     x x  + 1, --x x  + ----x x  + 8x x  + -x x x  + -x x x  + -x x x  +
      1 4      10 1 2    90  1 2     1 2   2 1 2 3   9 1 2 3   5 1 2 4  
     ------------------------------------------------------------------------
         2
     9x x x  + x x x x  + 1), {x , x })
       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     1             10                   1                    
o6 = (map(R,R,{-x  + -x  + x , x , --x  + x  + x , x  + -x  + x , x }), ideal
               3 1   9 2    5   1   7 1    2    4   1   7 2    3   2         
     ------------------------------------------------------------------------
      5 2   1               3  125 3     25 2 2   25 2        5   3  
     (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
      3 1   9 1 2    1 5    2   27 1 2   27 1 2    3 1 2 5   81 1 2  
     ------------------------------------------------------------------------
     10   2           2    1  4    1 3     1 2 2      3
     --x x x  + 5x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
      9 1 2 5     1 2 5   729 2   27 2 5   3 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                        
     {-10} | 295245x_1x_2x_5^6-36450x_2^9x_5-5x_2^9+164025x
     {-9}  | 15x_1x_2^2x_5^3-492075x_1x_2x_5^5+135x_1x_2x_5
     {-9}  | 15x_1x_2^3+492075x_1x_2^2x_5^2+270x_1x_2^2x_5+
     {-3}  | 15x_1^2+x_1x_2+9x_1x_5-9x_2^3                 
     ------------------------------------------------------------------------
                                                                       
     _2^8x_5^2+45x_2^8x_5-492075x_2^7x_5^3-405x_2^7x_5^2+3645x_2^6x_5^3
     ^4+60750x_2^9-273375x_2^8x_5-25x_2^8+820125x_2^7x_5^2+450x_2^7x_5-
     39226324511250x_1x_2x_5^5-5380840125x_1x_2x_5^4+2952450x_1x_2x_5^3
                                                                       
     ------------------------------------------------------------------------
                                                                             
     -32805x_2^5x_5^4+295245x_2^4x_5^5+19683x_2^2x_5^6+177147x_2x_5^7        
     6075x_2^6x_5^2+54675x_2^5x_5^3-492075x_2^4x_5^4+135x_2^4x_5^3+x_2^3x_5^3
     +1215x_1x_2x_5^2-4842756112500x_2^9+21792402506250x_2^8x_5+2989355625x_2
                                                                             
     ------------------------------------------------------------------------
                                                                  
                                                                  
     -32805x_2^2x_5^5+18x_2^2x_5^4-295245x_2x_5^6+81x_2x_5^5      
     ^8-65377207518750x_2^7x_5^2-44840334375x_2^7x_5+2460375x_2^7+
                                                                  
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     484275611250x_2^6x_5^2-66430125x_2^6x_5-18225x_2^6-4358480501250x_2^5x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^3+597871125x_2^5x_5^2+164025x_2^5x_5+135x_2^5+39226324511250x_2^4x_5^4-
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5380840125x_2^4x_5^3+2952450x_2^4x_5^2+1215x_2^4x_5+x_2^4+32805x_2^3x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2+27x_2^3x_5+2615088300750x_2^2x_5^5-358722675x_2^2x_5^4+492075x_2^2x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3+243x_2^2x_5^2+23535794706750x_2x_5^6-3228504075x_2x_5^5+1771470x_2x_5^
                                                                             
     ------------------------------------------------------------------------
                   |
                   |
                   |
     4+729x_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     8             4     9                      7 2   8    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                3 1   5 2    4   1  5 1   7 2    3   2           3 1   5 1 2
      -----------------------------------------------------------------------
                  16 3     524 2 2   72   3   4 2       8   2     4 2      
      + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      15 1 2   175 1 2   35 1 2   3 1 2 3   5 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      9   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

                5     4                  6                      12 2   4    
o16 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (--x  + -x x 
                7 1   9 2    4   1   1   5 2    3   2            7 1   9 1 2
      -----------------------------------------------------------------------
                  5 3     82 2 2    8   3   5 2       4   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   63 1 2   15 1 2   7 1 2 3   9 1 2 3    1 2 4  
      -----------------------------------------------------------------------
      6   2
      -x x x  + x x x x  + 1), {x , x })
      5 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  - 2x  + x , x , - 2x  + 2x  + x , x }), ideal (- x  -
                    1     2    4   1      1     2    3   2              1  
      -----------------------------------------------------------------------
                          3         3     2           2       2           2
      2x x  + x x  + 1, 4x x  - 4x x  - 2x x x  - 2x x x  - 2x x x  + 2x x x 
        1 2    1 4        1 2     1 2     1 2 3     1 2 3     1 2 4     1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :