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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

               1     9                                         6 2   9      
o3 = (map(R,R,{-x  + -x  + x , x , x  + 3x  + x , x }), ideal (-x  + -x x  +
               5 1   5 2    4   1   1     2    3   2           5 1   5 1 2  
     ------------------------------------------------------------------------
               1 3     12 2 2   27   3   1 2       9   2      2           2
     x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + x x x  + 3x x x 
      1 4      5 1 2    5 1 2    5 1 2   5 1 2 3   5 1 2 3    1 2 4     1 2 4
     ------------------------------------------------------------------------
     + x x x x  + 1), {x , x })
        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     6             5     1         4                    
o6 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , -x  + 3x  + x , x }),
               6 1   7 2    5   1  6 1   4 2    4  9 1     2    3   2   
     ------------------------------------------------------------------------
            1 2   6               3   1  3      1 2 2    1 2       18   3  
     ideal (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
            6 1   7 1 2    1 5    2  216 1 2   14 1 2   12 1 2 5   49 1 2  
     ------------------------------------------------------------------------
     6   2     1     2   216 4   108 3     18 2 2      3
     -x x x  + -x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
     7 1 2 5   2 1 2 5   343 2    49 2 5    7 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                
     {-10} | 100842x_1x_2x_5^6-74088x_2^9x_5-46656x
     {-9}  | 108864x_1x_2^2x_5^3-100842x_1x_2x_5^5+
     {-9}  | 182849716224x_1x_2^3+169375836672x_1x_
     {-3}  | 7x_1^2+36x_1x_2+42x_1x_5-42x_2^3      
     ------------------------------------------------------------------------
                                                        
     _2^9+43218x_2^8x_5^2+54432x_2^8x_5-16807x_2^7x_5^3-
     127008x_1x_2x_5^4+74088x_2^9-43218x_2^8x_5-18144x_2
     2^2x_5^2+426649337856x_1x_2^2x_5+83047723206x_1x_2x
                                                        
     ------------------------------------------------------------------------
                                                                             
     63504x_2^7x_5^2+74088x_2^6x_5^3-86436x_2^5x_5^4+100842x_2^4x_5^5+518616x
     ^8+16807x_2^7x_5^2+42336x_2^7x_5-74088x_2^6x_5^2+86436x_2^5x_5^3-100842x
     _5^5-52298274672x_1x_2x_5^4+131736761856x_1x_2x_5^3+248878780416x_1x_2x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _2^2x_5^6+605052x_2x_5^7                                                
     _2^4x_5^4+127008x_2^4x_5^3+559872x_2^3x_5^3-518616x_2^2x_5^5+1306368x_2^
     5^2-61014653784x_2^9+35591881374x_2^8x_5+22413546288x_2^8-13841287201x_2
                                                                             
     ------------------------------------------------------------------------
                                                                        
                                                                        
     2x_5^4-605052x_2x_5^6+762048x_2x_5^5                               
     ^7x_5^2-43581895560x_2^7x_5+10978063488x_2^7+61014653784x_2^6x_5^2-
                                                                        
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     38423222208x_2^6x_5-48393096192x_2^6-71183762748x_2^5x_5^3+44827092576x_
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     2^5x_5^2+56458612224x_2^5x_5+213324668928x_2^5+83047723206x_2^4x_5^4-
                                                                          
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     52298274672x_2^4x_5^3+131736761856x_2^4x_5^2+248878780416x_2^4x_5+
                                                                       
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     940369969152x_2^4+871075731456x_2^3x_5^2+3291294892032x_2^3x_5+
                                                                    
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     427102576488x_2^2x_5^5-268962555456x_2^2x_5^4+1693758366720x_2^2x_5^3+
                                                                           
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     3839844040704x_2^2x_5^2+498286339236x_2x_5^6-313789648032x_2x_5^5+
                                                                       
     ------------------------------------------------------------------------
                                                |
                                                |
                                                |
     790420571136x_2x_5^4+1493272682496x_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     7                   7                      8 2   7    
o13 = (map(R,R,{-x  + -x  + x , x , 2x  + -x  + x , x }), ideal (-x  + -x x 
                7 1   8 2    4   1    1   6 2    3   2           7 1   8 1 2
      -----------------------------------------------------------------------
                  2 3     23 2 2   49   3   1 2       7   2       2      
      + x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + 2x x x  +
         1 4      7 1 2   12 1 2   48 1 2   7 1 2 3   8 1 2 3     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

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     1             1     3                      4 2   1    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                3 1   3 2    4   1  8 1   5 2    3   2           3 1   3 1 2
      -----------------------------------------------------------------------
                   1 3      29 2 2   1   3   1 2       1   2     1 2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      24 1 2   120 1 2   5 1 2   3 1 2 3   3 1 2 3   8 1 2 4  
      -----------------------------------------------------------------------
      3   2
      -x x x  + x x x x  + 1), {x , x })
      5 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
--trying with basis element limit: 20

                                                                         2  
o19 = (map(R,R,{- 3x  - 2x  + x , x , - 3x  + 2x  + x , x }), ideal (- 2x  -
                    1     2    4   1      1     2    3   2               1  
      -----------------------------------------------------------------------
                          3         3     2           2       2           2
      2x x  + x x  + 1, 9x x  - 4x x  - 3x x x  - 2x x x  - 3x x x  + 2x x x 
        1 2    1 4        1 2     1 2     1 2 3     1 2 3     1 2 4     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 :