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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     9             1     10                      16 2   9    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (--x  + -x x 
               9 1   8 2    4   1  3 1    7 2    3   2            9 1   8 1 2
     ------------------------------------------------------------------------
                  7 3     107 2 2   45   3   7 2       9   2     1 2      
     + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
        1 4      27 1 2    72 1 2   28 1 2   9 1 2 3   8 1 2 3   3 1 2 4  
     ------------------------------------------------------------------------
     10   2
     --x x x  + x x x x  + 1), {x , x })
      7 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

               10     5             4     2         1     9              
o6 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , -x  + -x  + x , x }),
                7 1   2 2    5   1  9 1   5 2    4  6 1   4 2    3   2   
     ------------------------------------------------------------------------
            10 2   5               3  1000 3     750 2 2   300 2      
     ideal (--x  + -x x  + x x  - x , ----x x  + ---x x  + ---x x x  +
             7 1   2 1 2    1 5    2   343 1 2    49 1 2    49 1 2 5  
     ------------------------------------------------------------------------
     375   3   150   2     30     2   125 4   75 3     15 2 2      3
     ---x x  + ---x x x  + --x x x  + ---x  + --x x  + --x x  + x x ), {x ,
      14 1 2    7  1 2 5    7 1 2 5    8  2    4 2 5    2 2 5    2 5     5 
     ------------------------------------------------------------------------
     x , x })
      4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 1120x_1x_2x_5^6-60000x_2^9x_5-109375x_2^9+12000x_2^8x_5^2+43750x
     {-9}  | 122500x_1x_2^2x_5^3-13440x_1x_2x_5^5+49000x_1x_2x_5^4+720000x_2^
     {-9}  | 18757812500x_1x_2^3+2058000000x_1x_2^2x_5^2+15006250000x_1x_2^2x
     {-3}  | 20x_1^2+35x_1x_2+14x_1x_5-14x_2^3                               
     ------------------------------------------------------------------------
                                                                 
     _2^8x_5-1600x_2^7x_5^3-17500x_2^7x_5^2+7000x_2^6x_5^3-2800x_
     9-144000x_2^8x_5-175000x_2^8+19200x_2^7x_5^2+140000x_2^7x_5-
     _5+41287680x_1x_2x_5^5-75264000x_1x_2x_5^4+548800000x_1x_2x_
                                                                 
     ------------------------------------------------------------------------
                                                                             
     2^5x_5^4+1120x_2^4x_5^5+1960x_2^2x_5^6+784x_2x_5^7                      
     84000x_2^6x_5^2+33600x_2^5x_5^3-13440x_2^4x_5^4+49000x_2^4x_5^3+214375x_
     5^3+3001250000x_1x_2x_5^2-2211840000x_2^9+442368000x_2^8x_5+806400000x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     2^3x_5^3-23520x_2^2x_5^5+171500x_2^2x_5^4-9408x_2x_5^6+34300x_2x_5^5    
     ^8-58982400x_2^7x_5^2-537600000x_2^7x_5+392000000x_2^7+258048000x_2^6x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2-470400000x_2^6x_5-1715000000x_2^6-103219200x_2^5x_5^3+188160000x_2^5x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5^2+686000000x_2^5x_5+7503125000x_2^5+41287680x_2^4x_5^4-75264000x_2^4x
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     _5^3+548800000x_2^4x_5^2+3001250000x_2^4x_5+32826171875x_2^4+3601500000x
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     _2^3x_5^2+39391406250x_2^3x_5+72253440x_2^2x_5^5-131712000x_2^2x_5^4+
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2401000000x_2^2x_5^3+15756562500x_2^2x_5^2+28901376x_2x_5^6-52684800x_2x
                                                                             
     ------------------------------------------------------------------------
                                               |
                                               |
                                               |
     _5^5+384160000x_2x_5^4+2100875000x_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

                      2             4     2                        2   2    
o13 = (map(R,R,{5x  + -x  + x , x , -x  + -x  + x , x }), ideal (6x  + -x x 
                  1   3 2    4   1  5 1   5 2    3   2             1   3 1 2
      -----------------------------------------------------------------------
                    3     38 2 2    4   3     2       2   2     4 2      
      + x x  + 1, 4x x  + --x x  + --x x  + 5x x x  + -x x x  + -x x x  +
         1 4        1 2   15 1 2   15 1 2     1 2 3   3 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      2   2
      -x x x  + x x x x  + 1), {x , x })
      5 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     5                      8 2        
o16 = (map(R,R,{-x  + 3x  + x , x , -x  + -x  + x , x }), ideal (-x  + 3x x 
                7 1     2    4   1  3 1   9 2    3   2           7 1     1 2
      -----------------------------------------------------------------------
                   5 3     320 2 2   5   3   1 2           2     5 2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + 3x x x  + -x x x  +
         1 4      21 1 2    63 1 2   3 1 2   7 1 2 3     1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      5   2
      -x x x  + x x x x  + 1), {x , x })
      9 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  + x  + x , x }), ideal (3x  - 2x x 
                  1     2    4   1      1    2    3   2             1     1 2
      -----------------------------------------------------------------------
                      3       2 2       3     2           2       2      
      + x x  + 1, - 4x x  + 6x x  - 2x x  + 2x 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
      x 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 :