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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

               8     5             5                           17 2   5      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , x }), ideal (--x  + -x x  +
               9 1   2 2    4   1  2 1    2    3   2            9 1   2 1 2  
     ------------------------------------------------------------------------
               20 3     257 2 2   5   3   8 2       5   2     5 2      
     x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
      1 4       9 1 2    36 1 2   2 1 2   9 1 2 3   2 1 2 3   2 1 2 4  
     ------------------------------------------------------------------------
        2
     x 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

               1                  8     2         10     1              
o6 = (map(R,R,{-x  + x  + x , x , -x  + -x  + x , --x  + -x  + x , x }),
               2 1    2    5   1  3 1   7 2    4   9 1   2 2    3   2   
     ------------------------------------------------------------------------
            1 2                  3  1 3     3 2 2   3 2       3   3       2  
     ideal (-x  + x x  + x x  - x , -x x  + -x x  + -x x x  + -x x  + 3x 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
     ------------------------------------------------------------------------
       3     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} | 2x_1x_2x_5^6-6x_2^9x_5-2x_2^9+3x_2^8x_5^2+2x_2^8x_5-x_2^7x_5^3-
     {-9}  | 4x_1x_2^2x_5^3-6x_1x_2x_5^5+4x_1x_2x_5^4+18x_2^9-9x_2^8x_5-2x_2
     {-9}  | 8x_1x_2^3+12x_1x_2^2x_5^2+16x_1x_2^2x_5+18x_1x_2x_5^5-6x_1x_2x_
     {-3}  | x_1^2+2x_1x_2+2x_1x_5-2x_2^3                                   
     ------------------------------------------------------------------------
                                                                             
     2x_2^7x_5^2+2x_2^6x_5^3-2x_2^5x_5^4+2x_2^4x_5^5+4x_2^2x_5^6+4x_2x_5^7   
     ^8+3x_2^7x_5^2+4x_2^7x_5-6x_2^6x_5^2+6x_2^5x_5^3-6x_2^4x_5^4+4x_2^4x_5^3
     5^4+8x_1x_2x_5^3+8x_1x_2x_5^2-54x_2^9+27x_2^8x_5+9x_2^8-9x_2^7x_5^2-15x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     +8x_2^3x_5^3-12x_2^2x_5^5+16x_2^2x_5^4-12x_2x_5^6+8x_2x_5^5             
     2^7x_5+2x_2^7+18x_2^6x_5^2-6x_2^6x_5-4x_2^6-18x_2^5x_5^3+6x_2^5x_5^2+4x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^5x_5+8x_2^5+18x_2^4x_5^4-6x_2^4x_5^3+8x_2^4x_5^2+8x_2^4x_5+16x_2^4+24x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _2^3x_5^2+48x_2^3x_5+36x_2^2x_5^5-12x_2^2x_5^4+40x_2^2x_5^3+48x_2^2x_5^2
                                                                             
     ------------------------------------------------------------------------
                                                  |
                                                  |
                                                  |
     +36x_2x_5^6-12x_2x_5^5+16x_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

                 1                                                11 2  
o13 = (map(R,R,{--x  + 5x  + x , x , 7x  + 2x  + x , x }), ideal (--x  +
                10 1     2    4   1    1     2    3   2           10 1  
      -----------------------------------------------------------------------
                         7 3     176 2 2        3    1 2           2    
      5x x  + x x  + 1, --x x  + ---x x  + 10x x  + --x x x  + 5x x x  +
        1 2    1 4      10 1 2    5  1 2      1 2   10 1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      7x x x  + 2x x x  + x x x x  + 1), {x , x })
        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     5             7                            5 2   5    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + 5x  + x , x }), ideal (-x  + -x x 
                3 1   4 2    4   1  5 1     2    3   2           3 1   4 1 2
      -----------------------------------------------------------------------
                  14 3     61 2 2   25   3   2 2       5   2     7 2      
      + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      15 1 2   12 1 2    4 1 2   3 1 2 3   4 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
          2
      5x x x  + x x x x  + 1), {x , x })
        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,{x  + x , x , 6x  + x , x }), ideal (x  + x x  + x x  + 1,
                 2    4   1    2    3   2            1    1 2    1 4     
      -----------------------------------------------------------------------
          3      2         2
      6x x  + x x x  + 6x 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 :