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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

               9     9             1     1                      13 2   9                 9 3     27 2 2   9   3   9 2      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x  + x x  + 1, -x x  + --x x  + -x x  + -x x x  +
               4 1   2 2    4   1  2 1   2 2    3   2            4 1   2 1 2    1 4      8 1 2    8 1 2   4 1 2   4 1 2 3  
     ----------------------------------------------------------------------------------------------------------------------------
     9   2     1 2       1   2
     -x x x  + -x x x  + -x x x  + x x x x  + 1), {x , x })
     2 1 2 3   2 1 2 4   2 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

               3                  7     3         7     3                      3 2                  3  27 3     27 2 2   27 2    
o6 = (map(R,R,{-x  + x  + x , x , -x  + -x  + x , -x  + -x  + x , x }), ideal (-x  + x x  + x x  - x , --x x  + --x x  + --x x x 
               4 1    2    5   1  4 1   4 2    4  9 1   2 2    3   2           4 1    1 2    1 5    2  64 1 2   16 1 2   16 1 2 5
     ----------------------------------------------------------------------------------------------------------------------------
       9   3   9   2     9     2    4     3       2 2      3
     + -x x  + -x x x  + -x x x  + x  + 3x x  + 3x x  + x x ), {x , x , x })
       4 1 2   2 1 2 5   4 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} | 12x_1x_2x_5^6-54x_2^9x_5-12x_2^9+27x_2^8x_5^2+12x_2^8x_5-9x_2^7x_5^3-12x_2^7x_5^2+12x_2^6x_5^3-12x_2^5x_5^4+12x_2^4x
     {-9}  | 48x_1x_2^2x_5^3-108x_1x_2x_5^5+48x_1x_2x_5^4+486x_2^9-243x_2^8x_5-36x_2^8+81x_2^7x_5^2+72x_2^7x_5-108x_2^6x_5^2+108x
     {-9}  | 384x_1x_2^3+864x_1x_2^2x_5^2+768x_1x_2^2x_5+2916x_1x_2x_5^5-648x_1x_2x_5^4+576x_1x_2x_5^3+384x_1x_2x_5^2-13122x_2^9+
     {-3}  | 3x_1^2+4x_1x_2+4x_1x_5-4x_2^3                                                                                       
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                                                                 
     _5^5+16x_2^2x_5^6+16x_2x_5^7                                                                                                
     _2^5x_5^3-108x_2^4x_5^4+48x_2^4x_5^3+64x_2^3x_5^3-144x_2^2x_5^5+128x_2^2x_5^4-144x_2x_5^6+64x_2x_5^5                        
     6561x_2^8x_5+1458x_2^8-2187x_2^7x_5^2-2430x_2^7x_5+216x_2^7+2916x_2^6x_5^2-648x_2^6x_5-288x_2^6-2916x_2^5x_5^3+648x_2^5x_5^2
                                                                                                                                 
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                                                                 
                                                                                                                                 
                                                                                                                                 
     +288x_2^5x_5+384x_2^5+2916x_2^4x_5^4-648x_2^4x_5^3+576x_2^4x_5^2+384x_2^4x_5+512x_2^4+1152x_2^3x_5^2+1536x_2^3x_5+3888x_2^2x
                                                                                                                                 
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                                       |
                                                                                                       |
                                                                                                       |
     _5^5-864x_2^2x_5^4+1920x_2^2x_5^3+1536x_2^2x_5^2+3888x_2x_5^6-864x_2x_5^5+768x_2x_5^4+512x_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

                       7             3     7                        2    7                   3     499 2 2   49   3     2      
o13 = (map(R,R,{7x  + --x  + x , x , -x  + -x  + x , x }), ideal (8x  + --x x  + x x  + 1, 3x x  + ---x x  + --x x  + 7x x x  +
                  1   10 2    4   1  7 1   3 2    3   2             1   10 1 2    1 4        1 2    30 1 2   30 1 2     1 2 3  
      ---------------------------------------------------------------------------------------------------------------------------
       7   2     3 2       7   2
      --x x x  + -x x x  + -x x x  + x x x x  + 1), {x , x })
      10 1 2 3   7 1 2 4   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     5                  6                      5 2   5                 2 3     43 2 2   15   3   2 2      
o16 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (-x  + -x x  + x x  + 1, -x x  + --x x  + --x x  + -x x x  +
                3 1   2 2    4   1   1   7 2    3   2           3 1   2 1 2    1 4      3 1 2   14 1 2    7 1 2   3 1 2 3  
      ---------------------------------------------------------------------------------------------------------------------------
      5   2      2       6   2
      -x x x  + x x x  + -x x x  + x x x x  + 1), {x , x })
      2 1 2 3    1 2 4   7 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                      2 2      3     2          2        2
o19 = (map(R,R,{2x  - x  + x , x , x  + x , x }), ideal (3x  - x x  + x x  + 1, 2x x  - x x  + 2x x x  - x x x  + x x x  +
                  1    2    4   1   2    3   2             1    1 2    1 4        1 2    1 2     1 2 3    1 2 3    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 :

For the programmer

The object noetherNormalization is a method function with options.