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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

               5     2                    7                      12 2   2    
o3 = (map(R,R,{-x  + -x  + x , x , 5x  + --x  + x , x }), ideal (--x  + -x x 
               7 1   3 2    4   1    1   10 2    3   2            7 1   3 1 2
     ------------------------------------------------------------------------
                 25 3     23 2 2    7   3   5 2       2   2       2      
     + x x  + 1, --x x  + --x x  + --x x  + -x x x  + -x x x  + 5x x x  +
        1 4       7 1 2    6 1 2   15 1 2   7 1 2 3   3 1 2 3     1 2 4  
     ------------------------------------------------------------------------
      7   2
     --x x x  + x x x x  + 1), {x , x })
     10 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

               2                        7              7                    
o6 = (map(R,R,{-x  + x  + x , x , 9x  + -x  + x , x  + -x  + x , x }), ideal
               7 1    2    5   1    1   6 2    4   1   5 2    3   2         
     ------------------------------------------------------------------------
      2 2                  3   8  3     12 2 2   12 2       6   3   12   2  
     (-x  + x x  + x x  - x , ---x x  + --x x  + --x x x  + -x x  + --x x x 
      7 1    1 2    1 5    2  343 1 2   49 1 2   49 1 2 5   7 1 2    7 1 2 5
     ------------------------------------------------------------------------
       6     2    4     3       2 2      3
     + -x x x  + x  + 3x x  + 3x x  + x x ), {x , x , x })
       7 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} | 14x_1x_2x_5^6-24x_2^9x_5-14x_2^9+12x_2^8x_5^2+14x_2^8x_5-4x_2^
     {-9}  | 98x_1x_2^2x_5^3-84x_1x_2x_5^5+98x_1x_2x_5^4+144x_2^9-72x_2^8x_
     {-9}  | 4802x_1x_2^3+4116x_1x_2^2x_5^2+9604x_1x_2^2x_5+2016x_1x_2x_5^5
     {-3}  | 2x_1^2+7x_1x_2+7x_1x_5-7x_2^3                                 
     ------------------------------------------------------------------------
                                                                             
     7x_5^3-14x_2^7x_5^2+14x_2^6x_5^3-14x_2^5x_5^4+14x_2^4x_5^5+49x_2^2x_5^6+
     5-28x_2^8+24x_2^7x_5^2+56x_2^7x_5-84x_2^6x_5^2+84x_2^5x_5^3-84x_2^4x_5^4
     -1176x_1x_2x_5^4+2744x_1x_2x_5^3+4802x_1x_2x_5^2-3456x_2^9+1728x_2^8x_5+
                                                                             
     ------------------------------------------------------------------------
                                                                             
     49x_2x_5^7                                                              
     +98x_2^4x_5^3+343x_2^3x_5^3-294x_2^2x_5^5+686x_2^2x_5^4-294x_2x_5^6+343x
     1008x_2^8-576x_2^7x_5^2-1680x_2^7x_5+392x_2^7+2016x_2^6x_5^2-1176x_2^6x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     _2x_5^5                                                                 
     5-1372x_2^6-2016x_2^5x_5^3+1176x_2^5x_5^2+1372x_2^5x_5+4802x_2^5+2016x_2
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     ^4x_5^4-1176x_2^4x_5^3+2744x_2^4x_5^2+4802x_2^4x_5+16807x_2^4+14406x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     3x_5^2+50421x_2^3x_5+7056x_2^2x_5^5-4116x_2^2x_5^4+24010x_2^2x_5^3+
                                                                        
     ------------------------------------------------------------------------
                                                                          |
                                                                          |
                                                                          |
     50421x_2^2x_5^2+7056x_2x_5^6-4116x_2x_5^5+9604x_2x_5^4+16807x_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

                5                  4                            13 2         
o13 = (map(R,R,{-x  + x  + x , x , -x  + 2x  + x , x }), ideal (--x  + x x  +
                8 1    2    4   1  3 1     2    3   2            8 1    1 2  
      -----------------------------------------------------------------------
                5 3     31 2 2       3   5 2          2     4 2           2
      x x  + 1, -x x  + --x x  + 2x x  + -x x x  + x x x  + -x x x  + 2x x x 
       1 4      6 1 2   12 1 2     1 2   8 1 2 3    1 2 3   3 1 2 4     1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         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                         2         
o16 = (map(R,R,{10x  + x  + x , x , x  + -x  + x , x }), ideal (11x  + x x  +
                   1    2    4   1   1   4 2    3   2              1    1 2  
      -----------------------------------------------------------------------
                   3     7 2 2   1   3      2          2      2       1   2
      x x  + 1, 10x x  + -x x  + -x x  + 10x x x  + x x x  + x x x  + -x x x 
       1 4         1 2   2 1 2   4 1 2      1 2 3    1 2 3    1 2 4   4 1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         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

                                                                      2  
o19 = (map(R,R,{- 2x  + 2x  + x , x , 6x  - 4x  + x , x }), ideal (- x  +
                    1     2    4   1    1     2    3   2              1  
      -----------------------------------------------------------------------
                             3        2 2       3     2           2    
      2x x  + x x  + 1, - 12x x  + 20x x  - 8x x  - 2x x x  + 2x x x  +
        1 2    1 4           1 2      1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      6x x x  - 4x x x  + x x x x  + 1), {x , x })
        1 2 4     1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :