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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     8                        2          
o3 = (map(R,R,{5x  + 2x  + x , x , -x  + -x  + x , x }), ideal (6x  + 2x x  +
                 1     2    4   1  6 1   5 2    3   2             1     1 2  
     ------------------------------------------------------------------------
               35 3     31 2 2   16   3     2           2     7 2      
     x x  + 1, --x x  + --x x  + --x x  + 5x x x  + 2x x x  + -x x x  +
      1 4       6 1 2    3 1 2    5 1 2     1 2 3     1 2 3   6 1 2 4  
     ------------------------------------------------------------------------
     8   2
     -x x x  + x x x x  + 1), {x , x })
     5 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     7             4     5                                   
o6 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x  + 3x  + x , x }), ideal
               5 1   6 2    5   1  9 1   8 2    4   1     2    3   2         
     ------------------------------------------------------------------------
      1 2   7               3   1  3      7 2 2    3 2       49   3   7   2  
     (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  + -x x x 
      5 1   6 1 2    1 5    2  125 1 2   50 1 2   25 1 2 5   60 1 2   5 1 2 5
     ------------------------------------------------------------------------
       3     2   343 4   49 3     7 2 2      3
     + -x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
       5 1 2 5   216 2   12 2 5   2 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 38880x_1x_2x_5^6-63504x_2^9x_5-84035x_2^9+27216x_2^8x_5^2+72030x
     {-9}  | 360150x_1x_2^2x_5^3-116640x_1x_2x_5^5+308700x_1x_2x_5^4+190512x_
     {-9}  | 7414975286250x_1x_2^3+2401451388000x_1x_2^2x_5^2+12711386205000x
     {-3}  | 6x_1^2+35x_1x_2+30x_1x_5-30x_2^3                                
     ------------------------------------------------------------------------
                                                            
     _2^8x_5-7776x_2^7x_5^3-61740x_2^7x_5^2+52920x_2^6x_5^3-
     2^9-81648x_2^8x_5-72030x_2^8+23328x_2^7x_5^2+123480x_2^
     _1x_2^2x_5+195910410240x_1x_2x_5^5-259248729600x_1x_2x_
                                                            
     ------------------------------------------------------------------------
                                                                             
     45360x_2^5x_5^4+38880x_2^4x_5^5+226800x_2^2x_5^6+194400x_2x_5^7         
     7x_5-158760x_2^6x_5^2+136080x_2^5x_5^3-116640x_2^4x_5^4+308700x_2^4x_5^3
     5^4+1372257936000x_1x_2x_5^3+5447736945000x_1x_2x_5^2-319987003392x_2^9+
                                                                             
     ------------------------------------------------------------------------
                                                                  
                                                                  
     +2100875x_2^3x_5^3-680400x_2^2x_5^5+3601500x_2^2x_5^4-583200x
     137137287168x_2^8x_5+181474110720x_2^8-39182082048x_2^7x_5^2-
                                                                  
     ------------------------------------------------------------------------
                                                                   
                                                                   
     _2x_5^6+1543500x_2x_5^5                                       
     259248729600x_2^7x_5+137225793600x_2^7+266655836160x_2^6x_5^2-
                                                                   
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     352866326400x_2^6x_5-933897762000x_2^6-228562145280x_2^5x_5^3+
                                                                   
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     302456851200x_2^5x_5^2+800483796000x_2^5x_5+6355693102500x_2^5+
                                                                    
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     195910410240x_2^4x_5^4-259248729600x_2^4x_5^3+1372257936000x_2^4x_5^2+
                                                                           
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     5447736945000x_2^4x_5+43254022503125x_2^4+14008466430000x_2^3x_5^2+
                                                                        
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     111224629293750x_2^3x_5+1142810726400x_2^2x_5^5-1512284256000x_2^2x_5^4+
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     20012094900000x_2^2x_5^3+95335396537500x_2^2x_5^2+979552051200x_2x_5^6-
                                                                            
     ------------------------------------------------------------------------
                                                                        |
                                                                        |
                                                                        |
     1296243648000x_2x_5^5+6861289680000x_2x_5^4+27238684725000x_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

                9     2             7      3                      17 2  
o13 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (--x  +
                8 1   9 2    4   1  8 1   10 2    3   2            8 1  
      -----------------------------------------------------------------------
      2                 63 3     383 2 2    1   3   9 2       2   2    
      -x x  + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  +
      9 1 2    1 4      64 1 2   720 1 2   15 1 2   8 1 2 3   9 1 2 3  
      -----------------------------------------------------------------------
      7 2        3   2
      -x x x  + --x x x  + x x x x  + 1), {x , x })
      8 1 2 4   10 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

                6     4             7     7                      13 2   4    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                7 1   3 2    4   1  8 1   6 2    3   2            7 1   3 1 2
      -----------------------------------------------------------------------
                  3 3     13 2 2   14   3   6 2       4   2     7 2      
      + x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      4 1 2    6 1 2    9 1 2   7 1 2 3   3 1 2 3   8 1 2 4  
      -----------------------------------------------------------------------
      7   2
      -x x x  + x x x x  + 1), {x , x })
      6 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  + x , x , x  - x  + x , x }), ideal (x  + 2x x  + x x  +
                  2    4   1   1    2    3   2            1     1 2    1 4  
      -----------------------------------------------------------------------
           2 2       3       2      2          2
      1, 2x x  - 2x x  + 2x x x  + x x x  - x x x  + x x x x  + 1), {x , x })
           1 2     1 2     1 2 3    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 :