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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

                    7             1     5                        2   7      
o3 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , x }), ideal (2x  + -x x  +
                1   4 2    4   1  6 1   4 2    3   2             1   4 1 2  
     ------------------------------------------------------------------------
               1 3     37 2 2   35   3    2       7   2     1 2       5   2
     x x  + 1, -x x  + --x x  + --x x  + x x x  + -x x x  + -x x x  + -x x x 
      1 4      6 1 2   24 1 2   16 1 2    1 2 3   4 1 2 3   6 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

               5                  2                   3                    
o6 = (map(R,R,{-x  + x  + x , x , -x  + x  + x , x  + -x  + x , x }), ideal
               2 1    2    5   1  3 1    2    4   1   2 2    3   2         
     ------------------------------------------------------------------------
      5 2                  3  125 3     75 2 2   75 2       15   3        2  
     (-x  + x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  + 15x x x 
      2 1    1 2    1 5    2   8  1 2    4 1 2    4 1 2 5    2 1 2      1 2 5
     ------------------------------------------------------------------------
       15     2    4     3       2 2      3
     + --x x x  + x  + 3x x  + 3x x  + x x ), {x , x , x })
        2 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} | 10x_1x_2x_5^6-150x_2^9x_5-10x_2^9+75x_2^8x_5^2+10x_2^8x_5-25x_2
     {-9}  | 20x_1x_2^2x_5^3-150x_1x_2x_5^5+20x_1x_2x_5^4+2250x_2^9-1125x_2^
     {-9}  | 40x_1x_2^3+300x_1x_2^2x_5^2+80x_1x_2^2x_5+11250x_1x_2x_5^5-750x
     {-3}  | 5x_1^2+2x_1x_2+2x_1x_5-2x_2^3                                  
     ------------------------------------------------------------------------
                                                                             
     ^7x_5^3-10x_2^7x_5^2+10x_2^6x_5^3-10x_2^5x_5^4+10x_2^4x_5^5+4x_2^2x_5^6+
     8x_5-50x_2^8+375x_2^7x_5^2+100x_2^7x_5-150x_2^6x_5^2+150x_2^5x_5^3-150x_
     _1x_2x_5^4+200x_1x_2x_5^3+40x_1x_2x_5^2-168750x_2^9+84375x_2^8x_5+5625x_
                                                                             
     ------------------------------------------------------------------------
                                                                           
     4x_2x_5^7                                                             
     2^4x_5^4+20x_2^4x_5^3+8x_2^3x_5^3-60x_2^2x_5^5+16x_2^2x_5^4-60x_2x_5^6
     2^8-28125x_2^7x_5^2-9375x_2^7x_5+250x_2^7+11250x_2^6x_5^2-750x_2^6x_5-
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
     +8x_2x_5^5                                                              
     100x_2^6-11250x_2^5x_5^3+750x_2^5x_5^2+100x_2^5x_5+40x_2^5+11250x_2^4x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^4-750x_2^4x_5^3+200x_2^4x_5^2+40x_2^4x_5+16x_2^4+120x_2^3x_5^2+48x_2^3x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5+4500x_2^2x_5^5-300x_2^2x_5^4+200x_2^2x_5^3+48x_2^2x_5^2+4500x_2x_5^6-
                                                                             
     ------------------------------------------------------------------------
                                       |
                                       |
                                       |
     300x_2x_5^5+80x_2x_5^4+16x_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

                4                   9     10                      13 2  
o13 = (map(R,R,{-x  + 9x  + x , x , -x  + --x  + x , x }), ideal (--x  +
                9 1     2    4   1  7 1    9 2    3   2            9 1  
      -----------------------------------------------------------------------
                        4 3     6841 2 2        3   4 2           2    
      9x x  + x x  + 1, -x x  + ----x x  + 10x x  + -x x x  + 9x x x  +
        1 2    1 4      7 1 2    567 1 2      1 2   9 1 2 3     1 2 3  
      -----------------------------------------------------------------------
      9 2       10   2
      -x x x  + --x x x  + x x x x  + 1), {x , x })
      7 1 2 4    9 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

                      1                   5                        2   1    
o16 = (map(R,R,{7x  + -x  + x , x , 4x  + -x  + x , x }), ideal (8x  + -x x 
                  1   4 2    4   1    1   4 2    3   2             1   4 1 2
      -----------------------------------------------------------------------
                     3     39 2 2    5   3     2       1   2       2      
      + x x  + 1, 28x x  + --x x  + --x x  + 7x x x  + -x x x  + 4x x x  +
         1 4         1 2    4 1 2   16 1 2     1 2 3   4 1 2 3     1 2 4  
      -----------------------------------------------------------------------
      5   2
      -x x x  + x x x x  + 1), {x , x })
      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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                  2          
o19 = (map(R,R,{x  + 2x  + x , x , 2x  + 6x  + x , x }), ideal (2x  + 2x x  +
                 1     2    4   1    1     2    3   2             1     1 2  
      -----------------------------------------------------------------------
                  3        2 2        3    2           2       2      
      x x  + 1, 2x x  + 10x x  + 12x x  + x x x  + 2x x x  + 2x x x  +
       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 :