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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                   1                           14 2          
o3 = (map(R,R,{-x  + 2x  + x , x , -x  + x  + x , x }), ideal (--x  + 2x x  +
               9 1     2    4   1  6 1    2    3   2            9 1     1 2  
     ------------------------------------------------------------------------
                5 3     8 2 2       3   5 2           2     1 2          2
     x x  + 1, --x x  + -x x  + 2x x  + -x x x  + 2x x x  + -x x x  + x x x 
      1 4      54 1 2   9 1 2     1 2   9 1 2 3     1 2 3   6 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

               2     4                  8         4     2                    
o6 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , -x  + -x  + x , x }), ideal
               5 1   7 2    5   1   1   3 2    4  7 1   3 2    3   2         
     ------------------------------------------------------------------------
      2 2   4               3   8  3      48 2 2   12 2        96   3  
     (-x  + -x x  + x x  - x , ---x x  + ---x x  + --x x x  + ---x x  +
      5 1   7 1 2    1 5    2  125 1 2   175 1 2   25 1 2 5   245 1 2  
     ------------------------------------------------------------------------
     48   2     6     2    64 4   48 3     12 2 2      3
     --x x x  + -x x x  + ---x  + --x x  + --x x  + x x ), {x , x , x })
     35 1 2 5   5 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} | 168070x_1x_2x_5^6-131712x_2^9x_5-10240x_2^9+115248x_2^8x_5^2+
     {-9}  | 44800x_1x_2^2x_5^3-504210x_1x_2x_5^5+78400x_1x_2x_5^4+395136x
     {-9}  | 2293760000x_1x_2^3+25815552000x_1x_2^2x_5^2+8028160000x_1x_2^
     {-3}  | 14x_1^2+20x_1x_2+35x_1x_5-35x_2^3                            
     ------------------------------------------------------------------------
                                                                            
     17920x_2^8x_5-67228x_2^7x_5^3-31360x_2^7x_5^2+54880x_2^6x_5^3-96040x_2^
     _2^9-345744x_2^8x_5-17920x_2^8+201684x_2^7x_5^2+62720x_2^7x_5-164640x_2
     2x_5+1245715848090x_1x_2x_5^5-96848656800x_1x_2x_5^4+30118144000x_1x_2x
                                                                            
     ------------------------------------------------------------------------
                                                                       
     5x_5^4+168070x_2^4x_5^5+240100x_2^2x_5^6+420175x_2x_5^7           
     ^6x_5^2+288120x_2^5x_5^3-504210x_2^4x_5^4+78400x_2^4x_5^3+64000x_2
     _5^3+7024640000x_1x_2x_5^2-976234460544x_2^9+854205152976x_2^8x_5+
                                                                       
     ------------------------------------------------------------------------
                                                                             
                                                                             
     ^3x_5^3-720300x_2^2x_5^5+224000x_2^2x_5^4-1260525x_2x_5^6+196000x_2x_5^5
     66410507520x_2^8-498286339236x_2^7x_5^2-193697313600x_2^7x_5+6023628800x
                                                                             
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     _2^7+406764358560x_2^6x_5^2-31624051200x_2^6x_5-4917248000x_2^6-
                                                                     
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     711837627480x_2^5x_5^3+55342089600x_2^5x_5^2+8605184000x_2^5x_5+
                                                                     
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     4014080000x_2^5+1245715848090x_2^4x_5^4-96848656800x_2^4x_5^3+
                                                                   
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     30118144000x_2^4x_5^2+7024640000x_2^4x_5+3276800000x_2^4+36879360000x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3x_5^2+17203200000x_2^3x_5+1779594068700x_2^2x_5^5-138355224000x_2^2x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     4+107564800000x_2^2x_5^3+30105600000x_2^2x_5^2+3114289620225x_2x_5^6-
                                                                          
     ------------------------------------------------------------------------
                                                                  |
                                                                  |
                                                                  |
     242121642000x_2x_5^5+75295360000x_2x_5^4+17561600000x_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

                8     8             1                           17 2   8    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , x }), ideal (--x  + -x x 
                9 1   3 2    4   1  3 1    2    3   2            9 1   3 1 2
      -----------------------------------------------------------------------
                   8 3     16 2 2   8   3   8 2       8   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    9 1 2   3 1 2   9 1 2 3   3 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
         2
      x x x  + x x x x  + 1), {x , x })
       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

                 3                    9     2                      13 2  
o16 = (map(R,R,{--x  + 4x  + x , x , --x  + -x  + x , x }), ideal (--x  +
                10 1     2    4   1  10 1   3 2    3   2           10 1  
      -----------------------------------------------------------------------
                         27 3     19 2 2   8   3    3 2           2    
      4x x  + x x  + 1, ---x x  + --x x  + -x x  + --x x x  + 4x x x  +
        1 2    1 4      100 1 2    5 1 2   3 1 2   10 1 2 3     1 2 3  
      -----------------------------------------------------------------------
       9 2       2   2
      --x x x  + -x x x  + x x x x  + 1), {x , x })
      10 1 2 4   3 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 :